recursal/Fanatic-Fandom huggingface dataset:
Elaine Tewksbury

**Elaine Tewksbury** is a recurring villainess from Seasons Two and Three of *Coach*.

She was played by Robin Strasser.

In her first appearance in Season Two's "Haven't I Slept With You Somewhere Before?," it was revealed that Elaine had a sexual tryst with Hayden Fox five years prior to the episode's events. The tryst occurred in Miami (at the Blue Lagoon Motel), and it ended badly for both of them. Not wanting to get in a committed relationship with Elaine, Hayden left her without calling, with the excuse being that he needed to get ice.

Elaine turned heel and became obsessed with getting revenge on Hayden for dumping her, and it was in the aforementioned episode that Elaine was hired as the President of Minnesota State University. Elaine's first act as President was actually firing Hayden, but after Hayden reluctantly threatens to publicly reveal the reasons for his firing, Elaine recants. In "Professor Doolittle," Elaine orders Hayden to teach a class on ancient history, knowing he hasn't taught a class in over a decade--even though it's part of his job description. Elaine insisted that it was not payback for Miami, despite describing every detail by memory. At episode's end, after nearly all of Hayden's class passes, Elaine wants him teaching more classes, and this time, she does admit that she's doing it for payback; embracing her villainous persona by boldly boasting that she is a spiteful and vindictive woman.

In Season Three's "The Day That Moses Came to Town," the evil Elaine confronted Hayden over running NBA legend Rick Barry away from the university while posing as Athletic Director Howard Burleigh, but that didn't deter her from her plan to make basketball a more prominent sport at the school, by attempting to bring in high-level coach Terrance Moses. To add slight insult to injury, Elaine ordered Hayden to recruit Moses himself, with Howard going along to keep an eye on him. Elaine's last appearance came later in the season in "The Iceman Goeth," when she approaches Hayden (who is still down over his break-up with Christine) at the Touchdown Club. After venting about their failed relationships, the two engage in another attempted tryst, and like their past one, it ended in the same fashion: with Hayden leaving her. An enraged Elaine crashed into Hayden's office holding an axe, but Hayden manages to calm her down. Elaine notices how broken up Hayden is over his relationship, and actually takes joy in seeing that he's "even more screwed up than she is."
Ann Ostilow

**Ann Ostilow** (Jennifer Van Dyck) is a hidden villainess from "Sick," episode 5.19 of *Law & Order: SVU* (airdate March 30, 2004). She is the wife of Jeremy Ostilow and the mother of siblings Jeremy Jr. ("JJ" for short) and Lisette.

Ann first appeared along with her husband when detectives Elliot Stabler and Olivia Benson entered the home due to JJ posting online messages about raping and murdering a 5-year-old girl, and threatening to kill any police officers that came to stop him. During a one-on-one confrontation with Benson, Ann claimed that JJ has no obsession with violence and, and in response to questions about JJ harming Lisette, Ann stated that JJ loved his sister and he would say so if asked. However, it was revealed that JJ had a history of having violent thoughts and anger issues over anything and anyone plaguing him--including Lisette, who revealed that JJ mutilated the private parts of several of his sister's dolls.

Stabler suspected that JJ had all the signs of a victim of sexual abuse, and after JJ was committed to a psychiatric hospital, he was interviewed by Dr. George Huang. JJ revealed the truth to Huang: he was molested by a man he only referred to as "Billy," while his parents covered it up and were paid off. "Billy" was revealed as William Tripley, the owner of Tripley Toys, where Jeremy once worked. The molestation took place three years prior to the events, when Billy was just nine, and afterwards, JJ informed his parents about what Billy had done to him. In a shocking response, however, Ann turned heel (along with her husband) and accepted a $4 million payout from Billy for their silence, as well as JJ's. The greedy parents covered up the act for three years and agreed to the deal that the only way they can keep their ill-gotten money was to make sure JJ never went public regarding his molestation.

Billy was brought to trial afterwards, but later in the episode, JJ was shown on video recanting his statement, leading to Benson and Stabler confronting his parents at their home. Upon seeing the detectives, the evil Ann lashed out at Benson, telling her to lower her voice regarding their actions and JJ's whereabouts. The confrontation ended with an angry Jeremy slamming the door in the detectives' faces, and regarding Jeremy, he was arrested and locked up when he refused to give details on his son's whereabouts. Billy ended up acquitted in not only JJ's case, but the case regarding April Hodges, due to the fact that April was used by her evil grandmother as part of a monetary scam. Regarding Ann, her fate was never revealed.
Mrs. Mannick

**Mrs. Mannick** is the unseen main villainess from "Services Rendered," episode 7.10 of *Alfred Hitchcock Presents* (airdate December 12, 1961).

Mrs. Mannick is the wife of Dr. Ralph Mannick, who was visited by a young man who was suffering from amnesia after being struck on his head at a construction site. The man had Dr. Mannick's address and a $1000 bill in his pocket, originally leading to the man thinking that he could be Ralph Mannick, only to be assured otherwise upon seeing Dr. Mannick himself.

During Dr. Mannick's attempts to help his patient, the man picked up a photo of a brunette woman and asked who she was, with Dr. Mannick stating that the woman was his wife. The man was left wondering if he had one of his own, after which he left the office. As he was just about to enter the elevator, his memory returned, after which he informed Dr. Mannick.

The man also revealed where he got the $1000 bill from: at some point, Mrs. Mannick gave the bill to the man as payment to have her husband killed, with the reveal being cemented when the man pulled out a scalpel. No motive was given for Mrs. Mannick's actions, and the episode ended with Hitchcock's narration revealing that the hitman was arrested; while the evil Mrs. Mannick also ended up arrested (off-screen) for masterminding her husband's murder.
Nia Jax

**Nia Jax** is a villainess currently competing in WWE. After competing in live events for months, Nia made her long anticipated debut on the October 14, 2015 episode of *NXT*, competing against Evie. Nia turned heel when she laughed evilly at her opponent, while talking trash not only to Evie, but to the fans while dominating and later defeating her. After racking up victories against Deonna Purrazzo and Carmella, Nia formed an alliance with Eva Marie (referring to themselves as the Power Alliance), which included being in her corner in her NXT Women's Championship match against Bayley. After Eva was defeated, the evil Nia attacked Bayley and took possession of the title. Nia's title opportunity took place at NXT Takeover: London on December 16, but she was defeated via submission by Bayley.

Nia's pursuit of the NXT Women's Championship continued when she went after Asuka, but she was defeated at NXT Takeover: The End. On July 19, Nia was drafted to the Raw brand, and since then, she has been defeating enhancement talents in squash matches.
Candice Michelle

**Candice Michelle** is a villainess best known for competing in WWE, where she is a former Women's Champion.

Candice's tenure as a villainous Diva began on August 22, 2005 when she and Torrie Wilson attacked Diva Search winner Ashley Massaro on Raw. After that, Candice and Torrie aligned with Victoria to form an evil trio known as "Vince's Devils." Along with Torrie and Victoria, Candice spent most of her heel tenure pushing around not only Ashley, but Trish Stratus and Mickie James as well.

Candice received her first opportunity at the WWE Women's Championship on February 27, 2006, but she was defeated by Trish Stratus. After the match, Candice slapped Torrie, showing signs of dissension between the group. The trio broke up after Candice's Playboy cover was released just a week later. At the cover unveiling on Raw, Candice attempted to force Torrie to admit that her cover was better than either of Torrie's past Playboy covers. When Torrie refused to do so, Candice gave the signal for Victoria to attack her, and Candice delivered her own attack towards Torrie afterwards. Candice was defeated by Torrie in a Playboy Pillow Fight at WrestleMania 22, but she continued acting as a villainess after the event in various backstage segments with Vince McMahon, as well as resuming her feud with Torrie.

On an edition of ECW on July 11, Candice appeared as Kelly Kelly's dance partner in that night's edition of *Kelly's Exposé.* Before their performance, the evil Candice seduced Kelly Kelly in a backstage segment to maintain her status as a heel. Candice's tenure as a villainess ended six days later. After competing as a babyface for nearly three years (which included a reign as WWE Women's Champion in 2007), Candice began slowly turning villainous while competing at house shows in early 2009. However, Candice was released by WWE months later, abandoning any possible plans for Candice to become a villainess once again.
Chyna

**Chyna** (1969-2016) was a villainess best known for competing in WWF/E. She debuted in the World Wrestling Federation as the heel bodyguard/manager to Triple H in 1997, and both would align with Shawn Michaels and Rick Rude to form the faction known as D-Generation X. After HBK was forced to retire in 1998, the faction continued on as villains, with Triple H as their leader, but all of them, Chyna included, would become babyfaces at the end of that year and heading into 1999.

One day after becoming the first woman in history to compete in the Royal Rumble Match, Chyna became a villainess and delivered a low blow to Triple H after he saved her from being decimated by The Corporation. She would become the newest member of The Corporation shortly after her betrayal, and after turning villainous, Chyna revealed that she joined the group for fame and money. At WrestleMania XV, the evil Chyna managed to lure Triple H to the Corporation when they both turned against X-Pac during his match against Shane McMahon. This put Chyna back in her original role as the villainous valet to Triple H, a role that came to an end when she started pursuing the Intercontinental Championship.

By 2000, Chyna had formed an alliance with Chris Jericho after they feuded over the IC Championship. On the day after WrestleMania 2000, however, Chyna turned heel and helped Eddie Guerrero defeat Jericho to win the European Championship. As to why she turned against Jericho, Chyna stated that she couldn't resist his "Latino Heat." At the start, both Chyna and Guerrero were a villainous couple, but due to their fan reaction, both became babyfaces within a few months.

Chyna went into acting after her controversial WWF departure in 2001. She played several villainous roles, including playing Amazonian heel Natasha Tripova in *Relic Hunter*, and con artist Mary Jo Ponder on *Sabrina, the Teenage Witch*. She also played main villainess Rex in 2007's *Illegal Aliens.*
Betty

File:FFVIIR wiki icon.png

ベティ (Beti?)

**Betty** is a minor character in *Final Fantasy VII Remake*. She is a young girl living in the Sector 7 Slums with her father.

## Story

**Spoiler warning: *Plot and/or ending details follow.*** *(Skip section)*
Cloud and Tifa find Betty on the Sector 7 slums playground during Cloud's endeavor to put his name out as a mercenary-for-hire. Betty needs help finding her lost friends, and upon further probing it turns out she means her cats. Cloud is not thrilled about looking for cats, but Betty assures they are friendly and tame.

 File:Lost Friendss found Betty and cats from FFVII Remake.jpg
Cloud looks for the cats in the places where Betty says they usually hang out—junction leading to the station, outside the Seventh Heaven bar, and in the neighborhood near to the beginner's hall (a woman alerts Cloud to a cat that is stuck)-but every time Cloud approaches the white cats, they run off. Cloud and Tifa return to Betty and relay the bad news. She is at first crestfallen, but soon the three cats come running and Betty says that they were spooked by a scary man (Cloud).

Prior to the destruction of Sector 7, Aerith finds a lone Betty and carries her to a secret passages that leads to safety, and with Marle's guidance, Betty escapes to Sector 6 with her father. The two reach Wall Market, where Betty works to cheer up fellow survivors through the use of a nearby jukebox's music.

***Spoilers end here.***
## Quests

### Lost Friends

Main article: Lost Friends
In Chapter 3, Betty asks Cloud to find her three cats.

### The Power of Music

Main article: The Power of Music
In Chapter 14, Betty is by the jukebox in Wall Market where the player must find three musical discs in the area and play them to cheer up the locals.

## References

1. ↑ Final Fantasy VII Remake Ultimania, p.036

ru:Бетти
Colonel Tia Karim

**Colonel Tia Karim** was the main villainess from "Death of the Doctor," a two-part episode from Season Four of *The Sarah Jane Adventures*. (airdates 25 and 26 October 2010).

She was played by Laila Rouass, who also appeared on *The Royals* as the villainous Rani.

Tia Karim was a high ranking UNIT officer who appeared in Part 1 of the episode to inform Sarah Jane that The Doctor had passed on. She stated that the Shansheeth (a vulture-like race) discovered the Doctor's body and brought it to Earth for a proper ceremony at Base 5 underneath Snowdon, where Sarah Jane attends with Rani and Clyde, despite being suspicious of the claim. Part 1 ended with the Shansheeth being revealed as participating in a sinister plan to capture Sarah Jane and Jo Jones (another of the Doctor's companions), and take their memories.

In the beginning of Part 2, Karim turned heel when she was revealed to be in league with the Shansheeth, as Clyde, Rani, and Santiago (Jo's grandson) learned Karim's villainous ruse while they were in the air vents. After her reveal, the evil Karim locked the trio in the vents and turned up the heat, with (possible) intent on killing them. Karim later had Sarah Jane and Jo captured and placed on the Memory Weave, where she details her and the Shansheeth's evil plans. They planned to recreate the TARDIS key by using Sarah Jane and Jo's memories, so they can use the TARDIS to prevent death all over the universe, doing so by interfering with timelines. As for Karim, she wanted to leave Earth in the TARDIS, stating that there's nothing left for her. The villainess and her cohorts where thwarted when The Doctor appeared with the trio and told Sarah Jane and Jo to regain their memories, which led to the Weave exploding and killing the villains.
Lynette Richmont

**Lynette Richmont** (Kelly Overton) is the main villainess from "Wasted," episode 2.12 of *CSI: NY* (airdate January 18, 2006). She is the wife of Paul Richmont, who entered NYPD and confessed to murdering Dr. Rachel Jeffries.

Paul placed his gun on Don Flack's desk and confessed to killing Jeffries, and during his interrogation with Stella Bonasera, Paul stated that he killed Jeffries due to the fact that she lied to him about curing his rare blood disease, adding that he used leeches for symbolism while looking in Jeffries' eyes as he shot her. However, Stella revealed a hole in Paul's claim:  Jeffries was shot in the back, meaning that he couldn't have killed her. Stella's first encounter with Lynette included the reveal that Jeffries was using leeches during her curing sessions with Paul--sessions that cost $3,000 each--while lambasting Jeffries for scamming the couple.

Paul was eliminated as a suspect when his gun was inspected, as three bullets were fired but the third was shot *after* the murder by Paul, revealing that he was covering for someone. Stella and Lindsay Monroe combed through the jewelry from Jeffries' possession and took out a locket with a photo of a little girl, who resembled Lynette when the age was progressed, revealing Lynette as Jeffries' killer. As it turned out, Lynette was enraged over Jeffries scamming them, and this led to Lynette storming into Jeffries' office--with Paul's gun in her purse. She lambasted the greedy villainess during an MRI and confronted her for lying to them, only for Jeffries to callously defend her actions. Lynette demanded something of hers back, but when Jeffries ripped off Lynette's locket and handed it to her, she threw it down on the ground and stormed out of the office.

Shortly after the confrontation, Lynette turned heel by shooting Jeffries to death; waiting in ambush in an alley before firing two shots in her back. After the murder, Lynette informed Paul that she killed Jeffries, leading to Paul firing a third shot on the ground to make sure he would take the fall--mainly due to him having months to live. Even with her husband's selfless act, the evil Lynette portrayed Paul as a cold-blooded killer in her interview with Stella, doing so to mask her role as a murderous villainess. In the episode's climax, Paul was informed by Flack that he was free to go, much to Paul's shock; all the while, Lynette was shown in handcuffs, arrested for Jeffries' murder and begging Paul to "tell them what happened."
Cynthia Dern

**Cynthia Dern** (aka **Allison Porter**, aka **Allison Goldman**) was the true main villainess from "Ghosts," episode 1.08 of *Castle*. She was part of an eco-terrorist group that consisted of Jared Swanstrum and Susan Mailer, and after the Exxon Valdez Oil Spill, the three of them colluded to plant a bomb on an oil tanker. Swanstrum built the bomb, while Cynthia and Susan planned to plant the bomb.

However, the ship's captain, Sam Pike, got on board after the bomb was placed, causing a snag in the plan. Susan wanted to go back and defuse the bomb, not wanting harm to come to Pike, but Cynthia wanted the plan to go through and escaped while Susan went back. The bomb went off, crippling Pike and (assumingly) killing Susan.

Cynthia's true evil colors showed themselves after the events, as while she went into hiding, she sent an anonymous tip to the FBI and turned in Swanstrum. She later changed her name to Allison Porter, taking the identity from a three-month old baby who died in 1963, and met and married Michael Goldman.

20 years later, Cynthia (who was facing financial problems) went public with her version of the events, working with ghost writer Lee Wax to pen a book about the bombing. She was paid a few hundred dollars a week, with a big jackpot if her book became a best-seller. Cynthia painted herself as a hero in her story and placed all of the blame on former cohort Susan; depicting Susan as a maniacal villainess hellbent on bombing the tanker. Cynthia also claimed that she wanted to save Pike and clashed with Susan's heel persona, leading to dissuade the "villainous" Susan (as Cynthia claimed) planting the bomb anyway and eventually perishing.

However, to Cynthia's surprise, Susan had survived the bombing, though she was facially scarred. Susan had learned about Cynthia's book and her twisted version of the events, and she came out of hiding and warned Cynthia in a phone conversation that she would turn herself in and reveal the entire truth. Needing to keep her ruse alive, the evil Cynthia plotted to kill Susan, first by inviting her to her hotel room for drinks. She spiked Susan's glass and then filled the bathtub with oil, planning to drown Susan. Her plan was to make it appear that the "villainous" Susan committed suicide out of guilt, which would guarantee sympathy for Cynthia and increase book sales, as well as mask Cynthia's authentic heel persona in the process. Susan turned down the wine, stating that she doesn't drink, and became suspicious of Cynthia's motives. After seeing the tub filled with oil. Susan realized Cynthia's evil plan and fought with her, ending with Cynthia knocked unconscious and later drowned in the tub by Susan.
Jane Caldicott

**Jane Caldicott** is a villainess from "No Good Deed", Season 10 episode 15 of *CSI: Miami* (airdate March 4, 2012). She is played by *Royal Pains* actress Reshma Shetty, whose real-life husband, actor and financer Deep Katdare played a villain in the Season 3 Finale, "10-7".

## Summary

Caldicott was the head of an accounting firm, which employed accountant Henry Duncan, who was found dead after being run over by a boat while out surfing.

The investigation ultimately led the detectives of the Miami Police Department to Henry's employment at Caldicott's company and their investigation led to a revelation: one of Caldicott's clients, construction site owner Bob Scherner, was building highly priced houses on land he bought cheaply--land which also contained carcinogenic soil. Caldicott was revealed to be aware of the toxic land Scherner was working on and conspired with him to keep his actions covered for monetary benefits.

Henry eventually uncovered Scherner's illegal activities and confronted him and Caldicott, blasting her for allowing Scherner to build houses on unsafe land. In response, Caldicott ordered Henry to return to his office and to stay quiet about Scherner's work. Caldicott was brought in for interrogation by Ryan Wolfe and Horatio Crane, during which Caldicott tried to deny knowledge of Scherner's work. But with her fingerprints on the files proving Scherner's deeds, sealing her reveal, Caldicott confessed to her complicity in the illegal building. She also adamantly denied killing Henry to cover her and Scherner's tracks, stating that she had planned on bribing Henry into silence while subtly fingering Scherner as the true killer. While Caldicott and Scherner were ultimately proven innocent in Henry's murder, they were arrested for their illegal activities as they were shown being taken into custody in one of the episode's final scenes.
Lana Lang (Lois & Clark)

**Lana Lang** is a villainess from episode 3.14 of *Lois & Clark*, "Tempus, Anyone?"

She was played by Emily Procter.

Just like in the comics, Lana Lang is portrayed as the girlfriend of young Clark Kent, and the episode has Lana referred to as Clark's former girlfriend. Lana turned heel when she appeared in Tempus' alternate universe as Clark's fiancee. In the alternate reality, Lana and her family took Clark in after his parents died, and they began dating and became engaged, while also knowing about Clark's powers.

Lois Lane is transported to the alternate universe by Tempus, where she learns that she is deceased in the universe and meets Lana, while expressing shock over the fact that she and Clark are engaged. Lana displays her arrogant heel persona when she berated Clark for using his powers in public to save someone. Lana's belief was that one man can't save everyone, and she feared that Clark would be locked up by the government if he was found out.

Lana also appeared to be jealous of Lois' appearance, especially when Clark was concerned. She displayed her anger when she caught Lois and Clark together, as Lois was making Clark's trademark Superman outfit for him. At that moment, the evil Lana gave Clark a choice: either her or Superman. Clark refused to give up saving people, which led to Lana breaking up with Clark.

Lana never appeared in the real universe in the episode, but Clark stated to Lois that he and Lana broke up on good terms, and they're still friends.
Leena Reyes

**Leena Reyes** is a recurring villainess from Season Two of *Sleepy Hollow*.

She was played by Sakina Jaffrey.

Leena Reyes first appeared in the season's second episode, "The Kindred," and reclaimed the sheriff's position, replacing Frank Irving, who was institutionalized after Season 1. Once she assumed the position, Leena scolded Abbie for being late on roll call, and then later stated that "We are going to bring sanity back to this town," showing that she meant business.

Later in the episode, Leena turned heel and arrested Jenny in police archives for smuggling firearms, and later placed her in jail. After the arrest, the evil Leena snarled at Abbie and Ichabod Crane, displaying her heel persona and her indifference to Jenny and Ichabod being a consultant, wanting Ichabod out of police work. Leena later attended Frank Irving's interrogation, and cemented herself as a villainess when she stated that keeping him institutionalized would be best.

In the episode "Root of All Evil", it is revealed that Leena knows Abbie and Jenny Mills' birth mom, Lori Roberts, when they were children cause she tried to kidnapped them. It was Leena's testimony that led to Abbie and Jenny's mother being institutionalized at Tarrytown Psychiatric Hospital. In the "Root of All Evil", Leena gave Abbie a Tarrytown file containing the truth about her mother. When Leena got the chance to speak with Jenny, she told Jenny that it would be best if she got herself together because she'd hate to see her end up like her mother. Afterward, Leena arrived during a robbery, where she shot a culprit dead to defend Abbie. When seeing Ichabod around headquarters again, Leena continued her villainous persona and showed her displeasure at his presence; stating that she looked up Ichabod and found no files on his existence. She also informed Ichabod that he could not return to the department until he provides some provable documents. Trying to provide funding for the department, Leena went out hunting with the state official, unaware of a possessed Jenny trying to murder her.

Leena redeemed herself in "Deliverance", when she acted on the tip given by Abbie to raid the warehouse belonging to the Hellfire Club. Very impressed by Ichabod's work, she stated for him to keep in touch. Another time, the police department started getting calls about suicides in Tarrytown Psychiatric Hospital, in fact, the deaths were caused by the ghost of Gina Lambert. Leena gave the case to Abbie, and allowed her full control. She admitted to trusting Abbie's instincts.
Distinct Right Triangle of an Incircle

## **Problem**

**Problem created by Tao Steven Zheng 鄭濤**

This is an original problem I created in 2015. I was inspired by a problem from the *Jiuzhang Suanshu*. See Jiuzhang Suanshu Gougu 16.

How many distinct right triangles with integer sides circumscribe a circle of radius 10? List all the triangles.

**Note:** By distinct I mean we do not count any permutation of

(
a
,
b
,
c
)

{\displaystyle (a, b, c)}

. For example, the Pythagorean triple

(
12
,
5
,
13
)

{\displaystyle (12, 5 ,13) }

 is the same triangle as

(
5
,
12
,
13
)

{\displaystyle (5, 12 ,13) }

.

## **Solution**

We use one of the formulas relating the radius of an incircle to the three sides of the right triangle

# r

a
b
−
c

2

{\displaystyle
r=\frac{a + b -c}{2}}

, where

r

{\displaystyle r\; }

 is the radius,

a
,
b

{\displaystyle a, b}

 are the two legs, and

c

{\displaystyle c\;}

 is the hypotenuse.

Substituting the radius gives

# 10

a
b
−
c

2

{\displaystyle 10 = \frac{a+b-c}{2} }

or alternatively

# 20
a
b
−
c
.

{\displaystyle 20 = a+b-c. }

Since

# c

a
2

b
2

{\displaystyle c = \sqrt{a^2 + b^2}}

 for right triangles, we get

# 20
a
b
−

a
2

b
2

.

{\displaystyle 20 = a + b - \sqrt{a^2 + b^2}. }

Subsequently, we have

a
b
−
20

=

a
2

b
2

(
a
b
)
−
20

=

(
a
b

)
2

−
2
a
b

(
a
b
)

2

−
40
(
a
b
)
400

=
(
a
b

)
2

−
2
a
b
.

{\displaystyle \begin{align}
a + b - 20 &= \sqrt{a^2 + b^2} \\[10 pt]
(a + b) - 20 &= \sqrt{(a+b)^2 - 2ab} \\[10 pt]
{(a + b)}^{2} - 40(a+b) + 400 &= (a+b)^2 - 2ab.
\end{align} }

Isolating

a

{\displaystyle a\;}

 yields

400

=
40
(
a
b
)
−
2
a
b

200

=
20
(
a
b
)
−
a
b

20
a
−
a
b

=
200
−
20
b

a
(
20
−
b
)

=
200
−
20
b

a

=

200
−
20
b

20
−
b

.

{\displaystyle \begin{align}
400 &= 40(a+b) - 2ab \\[10 pt]
200 &= 20(a+b) - ab \\[10 pt]
20a - ab &= 200 - 20b \\[10 pt]
a(20 - b) &= 200 - 20b \\[10 pt]
a &= \frac{200 - 20b}{20 - b}.
\end{align} }

Impose the condition

a
>
b

{\displaystyle a > b}

, so that we only need to check the values

21
≤
b
≤
34

{\displaystyle 21 \le b \le 34 }

.
The only integer solutions for

a
,
b

{\displaystyle a, b}

 are

(
220
,
21
)
,
(
120
,
22
)
,
(
70
,
24
)
,
(
60
,
25
)
,
(
45
,
28
)
,
(
40
,
30
)

{\displaystyle (220,21), (120,22), (70,24), (60,25), (45,28), (40,30) }

. Hence, there are 6 distinct triangles:

(
220
,
21
,
221
)

(
120
,
22
,
122
)

(
70
,
24
,
74
)

(
60
,
25
,
65
)

(
45
,
28
,
53
)

(
40
,
30
,
50
)
.

{\displaystyle \begin{align}
(220,21,221) \\[10 pt]
(120,22,122) \\[10 pt]
(70,24,74) \\[10 pt]
(60,25,65) \\[10 pt]
(45,28,53) \\[10 pt]
(40,30,50).
\end{align} }
Carolyn Morrow

**Carolyn Morrow** (Sarah Buxton) is a hidden villainess from "Flight Risk," episode 7.18 of *CSI: Miami* (airdate March 16, 2009). She is a Pathways Air flight attendant and a colleague of Suzanne Grady, the episode's victim.

Carolyn was introduced while she was cleaning up the cabin of the plane used to fly Pathways Flight 88, a scheduled domestic passenger flight from San Diego International Airport to Miami International Airport. As she was doing so, she met Eric Delko and Calleigh Duquesne. Brushing off Suzanne's death as merely being "a tragedy," Carolyn responded to their suspicions of disposing evidence by stating that she "didn't imagine that anything [in the cabin] mattered."

Regarding bruises on her forearms and legs, she resentfully voiced that they were "what you get when you get stuck in coach." Along with claiming she first met Suzanne when they boarded on the day of her death, Carolyn gave the trash bag she was using to the pair. Immediately after doing that, she openly admitted to her bitterness over "cost cuts" that left her a self-described "garbage collector" on top of her other jobs: "waitress," "safety instructor," and "shrink."

However, a photo showing her attending a party in San Diego on the day before Flight 88's departure, with Suzanne and several other people present. When Eric interrogated her about it, Carolyn unleashed the true extent of her jealousy by ranting about how Suzanne was rewarded for behavior which indicated she "wouldn't separate work and play" (in her mind) by "getting to work first class." Viewing her own work as punishment "for being good at [her] job," the spiteful woman decided to "help [Suzanne] enjoy herself a little bit more" at the aforementioned party by lacing a bottle of club soda with a large amount of an unknown alcoholic beverage before giving it to her under the guise of it being "just soda."

Carolyn's motive for spiking Suzanne's drink was her hope that she would subsequently miss Flight 88, which would have resulted in her being "bounced to the Minneapolis corridor," thus allowing for her to take her place aboard the long-haul flights she worked on and be paid more—which she firmly believed she was entitled to because of her "seniority" over her. After finishing her confession, Eric allowed for her to leave, but she remained a suspect; later on, a flashback showed her asking federal air marshal Aaron Nolan to help her with a woman in coach class aboard Flight 88 who she described as someone who "[wouldn't] calm down." Furthermore, her decision to orchestrate Suzanne's hangover was discovered as having ultimately allowed for another colleague of hers to drug and then indirectly kill her by accident.
Kara (Supergirl)

**Kara**is the titular protagonist in CBS' *Supergirl* series. Played by Melissa Benoist, Kara became a villainess in episode 1.16, "Falling."

While saving a fireman trapped by debris, Kara was inadvertently exposed to red kryptonite while removing the debris. Following the exposure, Kara slowly underwent a change in personality, beginning with wearing more stylish clothing to work. Her duties as Supergirl were also affected, as she fought off and defeated a Fort Rozz escapee, only to actually let him go. Supergirl's actions were caught on video by Siobhan Smythe, who wanted to give the video to Cat to run as a story. When Cat turns her down, Siobhan attempted to send the story to her former employers at the Daily Planet, but Kara--having heard everything--decided to send a copy of Siobhan's email to Cat, which led to Cat firing Siobhan. Kara continued turning villainous when she mocked Siobhan, and later on, she began coming on to James while at a club, doing so while trashing his current girlfriend, Lucy.

Kara turned heel during a conversation with Cat, when she--as Supergirl--actually held Cat over a balcony. She dropped Cat, but saved her just in time, while saying that she no longer wants to be a hero. Kara's heel turn was cemented when she appeared to Alex at their home in an all black Kryptonian outfit, similar to Astra and Non, and laughed evilly at Alex after using her lasers to burn her old outfits. Kara later lashed out at Alex, who told her sister that she's not herself. However, the evil Kara stated that she feels freer than she ever did, and as she looked out the window, she made a vow to make those who don't already worship her do so, and flew off.

Kara unleashed havoc on National City, even taking down Alex during her villainous rampage. Hank, as J'onn J'onzz, engaged the villainess as a distraction for Alex to fire Maxwell Lord's antidote towards Kara. Back to normal, Kara felt extreme remorse for her actions while under the spell of the red kryptonite.
Olivia Burtis

**Olivia Burtis** was the hidden main villainess from *Satan's School for Girls*(2000).

She was played by Kate Jackson

Olivia Burtis was the dean of the Fallbridge College for Girls, where Beth Hammersmith attended under the name Karen Oxford to find out the truth about her sister's death. Olivia turned heel when she was revealed as the leader of a satanic cult of witches known as "The Five," which consisted of students at the college. Olivia also revealed that she was involved in killing Beth's parents and sister, with the plan to coerce Beth to join her cult. After Beth took down the students, the evil Olivia faced Beth herself, all the while arrogantly stating that she wasn't powerful enough. However, Beth's power from killing The Five ended up being strong enough to kill Olivia, as a bolt of lightning crashed through a window and destroyed her.
Eva Marie

**Eva Marie** is a villainess currently competing in WWE. She debuted on *Raw* on July 22, 2013, as one of the two newbies on *Total Divas*, and delivered a slap to WWE Hall of Famer Jerry "The King" Lawler in an attempt to make a name for herself. She later aligned herself with the Bella Twins, mainly as part of Brie Bella's feud with Natalya. In 2014, Eva Marie was one of the many Divas enlisted by the evil Stephanie McMahon to punish Nikki Bella for her sister's actions towards Stephanie at the Payback event (Brie slapped Stephanie's face after she walked out of WWE).

After a lengthy hiatus and heavy training to improve her wrestling skills, Eva Marie made her return to action on the July 22, 2015 episode of NXT, where she defeated Cassie. As part of her new persona, Eva Marie entered donning a long red robe, which she would slowly remove once she enters the ring. In the weeks that followed, the evil Eva Marie picked up victories over numerous Divas, while later forming an alliance with the villainous Nia Jax. In November, Eva Marie received her first opportunity at the NXT Women's Championship, losing to Bayley despite Nia Jax's involvement. The villainous pair continued their feud with Bayley and Carmella, which included defeating the duo on the February 24, 2016 episode of NXT.

Eva returned to the main roster in the final week leading to WrestleMania 32, as a babyface by association with *Total Divas* castmates Brie Bella, Alicia Fox, Paige, and Natalya--as part of Brie's feud with Lana. At WrestleMania, Eva Marie was part of the victorious babyface side of a 10 woman tag team match, despite displaying her familiar villainous persona. On July 19, 2016, Eva Marie was drafted to the *SmackDown* roster as part of the returning Brand Extension. Eva turned heel on the following week's edition of *SmackDown*, when she upstaged the rest of the SmackDown women during their entrances. Eva Marie's heel persona included a voiceover introduction in her entrance, and in the following weeks, she was supposed to make her in-ring debut, only to create unfortunate events to get out of competing.
Dr. Sosa

**Dr. Sosa** was the tertiary villainess of the 2018 film *Dark Sister*.

She was played by Nicola Bartlett.

Dr. Sosa was introduced as the therapist of artist Cassie, helping her as she finds herself tormented by dreams depicting murders. In actuality, however, Dr. Sosa was the mother of both Cassie and her long-lost twin sister Diana (the film's main villainess). As revealed by Cassie's "father" Hector, Dr. Sosa had been researching human emotion and the possibility that one sentiment being having the power to control it. As part of her research, Dr. Sosa performed experiments on her daughters, among other patients; including Hector (who was not actually the father of her children) and his twin brother, Apollo.

As an infant, Cassie was taken from her mother's stranglehold by Hector fleeing with as a fire broke out in her home, leading Dr. Sosa to concentrate her experiments on Diana. The film's events have Diana having escaped her mother's clutches, turning heel by using her powers to become a serial killer. As Cassie began to realize her dreams were connecting her to the serial killer, Dr. Sosa began to reveal her true colors during a meeting with Cassie by trying to give her an injection, possibly intending to recapture Cassie for more experiments. After Hector and Cassie's attempt to use an emotion enhancement drug to defeat Diana left her bloodied and weak, Dr. Sosa abducted her daughter and brought her back to her lab, restraining her to a table.

The evil Dr. Sosa briefly torturing her daughter with electric shocks in punishment for her escape, while also threatening to continue using the torture if Diana didn't "behave". Dr. Sosa later told Diana that she intended to impregnate her with a child that would "change the world"; likely intending to use said child in more sinister experiments. Dr. Sosa then had Apollo have sex with the captive Diana, smoking a cigarette while coldly watching her daughter's rape and shocking her again after seeing her whisper to Apollo. But shortly afterwards, Hector and Cassie took over their respective twins' body, with Diana attacking her mother with a knife and tying her to her chair. After accusing Cassie of (in her mind) being a bad influence on Diana, Diana killed Apollo with a stab to the neck. Dr. Sosa then attempted to manipulate her daughters against each other, claiming Cassie was jealous of Diana and wouldn't give her the answers she wanted, while sarcastically claiming Diana was not at fault for her killing spree. Unconvinced, Diana went about killing her demented mother by strangling her to death with a wire.
Jacklyn Dussault

**Jacklyn Dussault** is a recurring villainess from Season Three of *Devious Maids*.

She was played by Michelle Hurd.

Jacklyn Dussault is the head of talent for Ducerna Records, and in her first appearance in Episode 3.07, "The Turning Point," she interviews Carmen Luna after one of her talent agents discovers her. Jacklyn is married to real estate agent, Sebastian Dussault, who--unbeknownst to Jacklyn--is dating Carmen, and when Carmen learns that Jacklyn is Sebastian's wife, she originally turns down her offer. After Jacklyn stated to Carmen in another meeting that she is not a star, because stars will do whatever it takes to be famous, Carmen changes her mind and signs with Jacklyn. Regarding the complicated situation, it is revealed that Jacklyn had Sebastian sign a prenuptial agreement that protected her finances in a divorce, and their affair could result in a divorce that could leave Sebastian broke.

In the following episode, "Cries and Whispers," Jacklyn invites Carmen on a double date, leading to Carmen asking Doug, a coffee house waiter who has a crush on Carmen, to pose as her boyfriend. During the date, Jacklyn becomes upset at Sebastian for flirting with a waitress, and later reveals his numerous other dalliances. Jacklyn hinted a heel turn in last scene of the episode, when she seductively made out with Carmen after they vented about the men in their lives. In the following episode, "Bad Girl," Jacklyn confirmed Carmen's suspicions: she has a crush on Carmen. Under Sebastian's insistence, Carmen goes to Jacklyn's home for a potential date, only for Carmen to turn down her advances later on and telling her to leave Sebastian because she deserved better. At the end of the episode, Jacklyn found Carmen's phone when she heard it ringing due to Sebastian attempting to call her, and it is at that moment that she figures out that Carmen is Sebastian's lover.

Jacklyn turned heel in the episode, "Whiplash," as she gazed at Carmen's photo and plotted revenge on her. She pretended to believe Carmen's story that Sebastian is her realtor and not her lover, keeping her heel persona hidden from Carmen. During Carmen's TV interview, a video of Carmen and Sebastian having sex was shown on a monitor, which was supposed to show her music video. After the shock, Carmen turns around to see Jacklyn smiling evilly at her, revealing herself as the culprit. Later in the episode, Carmen apologizes to Jacklyn and states that her relationship with Sebastian is over, but Jacklyn stated that Carmen's words meant nothing. To Carmen's surprise, Jacklyn told her that she will not be dropped, but she also stated that Carmen's record will not be released, saying that Carmen is under contract and her record will be put on hold for a long time. Subsequently, the evil Jacklyn snarled that Carmen can't make music for any other label, and that she owns her.
Liz Picket

**Liz Picket** (Geneva Carr) is the hidden villainess from *Person of Interest* episode 2.07, "Critical." She is a nurse who was assisting Dr. Madeline "Maddy" Enright (the episode's protagonist) in the operation of energy CEO Oliver Veldt. Liz turned heel when she was revealed to be a plant working for the episode's main villain Alistair Wesley, who was blackmailing Maddy to kill Veldt, threatening to kill Maddy's wife, Amy, if she didn't comply.

Liz's villainous turn came during the operation, as Maddy discovered that Veldt's artery was nicked by Liz intentionally. Liz stated to Maddy that she was the back-up plan to kill Veldt in case Maddy didn't comply, and it was also revealed that she leaked the info about Veldt's operation to Wesley. After the revelation, the evil Liz told Maddy to let Veldt die, and reached for a scalpel with intent to attack Maddy. Finch, who was planted as a doctor on Maddy's side, knocked out the villainess with an overhead light.
Eve Windmere

**Eve Windmere** is the hidden main villainess from the short story, *The Sinister Sheep Heist*.

Eve Windmere is the sheriff of a small rural farm town, which has been plagued by a thief stealing sheep. The town's deputy, Claire Farrell, had been on the case and visited Eve late one night, informing her that she had solved the case, despite Eve's orders to drop the case. Claire showed Eve a file containing her findings, with Claire confirming that the thefts were an inside job.

However, when Claire opened Eve's curtains, she was stunned to see a flock of sheep captive in a small pen outside Eve's house, while a full moon appeared in the sky. Eve turned heel by snarling at Claire and suddenly transforming into a werewolf, while reminding Claire of her warning to drop the case.

As the twist revealed, Eve was the villainess behind the thefts, doing so while prowling as a werewolf and intending to devour the sheep. Her attempt to divert Claire from the case was to keep her heel persona a secret, and after her transformation, the evil Eve killed Claire by slashing her throat with her claws, and gave a loud howl afterwards.
Julie Randall

**Julie Randall** (Dorion Davis) was a surprise hidden villainess from "Cold Case," episode 2.04 of *Sue Thomas: F.B.Eye* (airdate November 9, 2003). She was a 22-year old bank teller who was shot and killed during a robbery at her bank on August 23, 1991.

In the years that passed, Julie's mother, Phyllis Randall checked with the FBI regularly regarding updates on her daughter's case, and it was in the episode's events that Sue Thomas and Lucy Dotson visited Phyllis, who was revealed to be dying of cancer. Sue investigated the 12-year-old murder case, which included inspecting video footage of the robbery. She discovered that Julie had uttered something to the robber, but couldn't properly read Julie's lips due to the footage, leading to Tara Williams piecing together all of the security footage to get a clear view of Julie's dialogue. After the material was played, Sue was surprised to learn that Julie appeared to give words of warning to the robber regarding a guard being behind him, after which the robber shot and killed the guard. The exact words, "Look out behind you!," came from Julie's lips, leading Sue and others to the shocking reveal that Julie was a villainess instead of an innocent victim.

At some point during her time as a bank teller, Julie turned heel and collaborated in the robbery of her own bank, with her words of warning to her co-conspirator leading to the guard's death. As revealed by Julie's friend, Mitzi Wells, the evil Julie had engaged in an affair with married car salesman Rick Cooper, leading to the belief that Rick was Julie's cohort and killer, while suggesting that Julie's heel turn was motivated by the affair. However, it was uncovered that Rick had died five years prior to the events, and his widow revealed that she and Rick were overseas celebrating their fifth wedding anniversary at the time of the robbery. DNA on the mask worn in the robbery revealed that the true robber was Rick's brother, Brian Cooper, who shot and killed the guard after Julie's warning, and killed Julie out of fear that she would turn against him. With everyone agreeing that Phyllis could not know the truth about Julie's heel persona, a deal was made where Brian received 15 years for the robbery and for Julie's murder.