nyuuzyou/wikis huggingface dataset:
Palindromic Triangular Numbers/Mistake 1

== Source Work ==
:
:The Dictionary
::$15$
== Mistake ==
:There are $40$ palindromic triangular numbers below $10^7$.
== Correction ==
There are in fact $22$ of them below $10^7$.
The $23$rd one is $35 \, 133 \, 153$, which is $3 \cdotp 5 \times 10^7$.
In , this statement has been removed.
== Sources ==
* : $15$
Category:Mistakes/Curious and Interesting Numbers
Palindromic Triangular Numbers/Mistake 2

== Source Work ==
:
:The Dictionary
::$828,828$
== Mistake ==
:The only triangular palindrome, apart from $55$, $66$ and $666$.
== Correction ==
From Palindromic Triangular Numbers, it can be seen that the sequence of triangular numbers which are also palindromic begins:
:$1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15 \, 051, 66 \, 066, 617 \, 716, 828 \, 828, 1 \, 269 \, 621, 1 \, 680 \, 861, 3 \, 544 \, 453, \ldots$
In , this statement has been removed, and replace by Palindromic Triangular Numbers with Palindromic Halves, which is what he may have been searching for here.
== Sources ==
* : $828,828$
Category:Mistakes/Curious and Interesting Numbers
Palindromic Triangular Numbers whose Index is Palindromic

== Sequence of Palindromic Triangular Numbers with Palindromic Indices ==
The sequence of palindromic triangular numbers whose indices are also palindromic begins:
:$1, 3, 6, 66, 3003, 66 \, 066, 617 \, 716, 3 \, 544 \, 453, 6 \, 172 \, 882 \, 716, \ldots$
Their indices are:
:$1, 2, 3, 11, 77, 363, 1111, 2662, 111 \, 111, \ldots$
== Sources ==
* : $15$
* : $15$
Category:Triangular Numbers
Category:Palindromic Numbers
Palindromic Triangular Numbers with Palindromic Halves

== Sequence of Palindromic Triangular Numbers ==
The sequence of palindromic triangular numbers which can be split into palindromic halves consists of:
:$55, 66, 828 \, 828$
There may be others, but none are known.
== Sources ==
* : $828,828$
Category:Triangular Numbers
Category:Palindromic Numbers
Palindromic Triangular Numbers with Palindromic Index/Historical Note

== Historical Note on Palindromic Triangular Numbers with Palindromic Index ==
 reports in that reports on this sequence (in particular $363 \, 474 \, 363$) in , Volume $24$, page $184$, but this has not been corroborated.
== Sources ==
* : $363,474,363$
Category:Historical Notes
Pandiagonal Magic Square/Examples/Order 4

== Example of Order $4$ Pandiagonal Magic Square ==
This is an example of an order $4$ pandiagonal magic square:
:$\begin {array} {|c|c|c|c|} \hline 1 & 8 & 11 & 14 \\ \hline 12 & 13 & 2 & 7 \\ \hline 6 & 3 & 16 & 9 \\ \hline 15 & 10 & 5 & 4 \\ \hline \end {array}$
It is noted by inspection that:
:the sum of the elements of each row is $34$
:the sum of the elements in each column is $34$
:the sum of the elements along each diagonal is $34$
:the sum of the elements along each broken diagonal, for example $11, 7, 6, 10$, is $34$
== Sources ==
* 
Category:Pandiagonal Magic Squares
Pandigital Integer Formed by Digits in Alphabetical Order

== Theorem ==
The number $8 \, 549 \, 176 \, 320$ is the pandigital integer formed from the digits from $0$ to $9$ arranged in alphabetical order.
== Proof ==
== Sources ==
* : $8,549,176,320$
Category:Pandigital Integers
Category:8,549,176,320
Pandigital Integers remaining Pandigital on Multiplication

== Theorem ==
Certain pandigital integers remain pandigital when multiplying them by certain single-digit integers:
The sequence:
:$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$
contains all pandigital integers with at least $4$ nontrivial pandigital multiples, of which:
:$1098765432, 1234567890$
has $5$.
== Sources ==
* : $123,456,789$
* : $123,456,789$
Category:Pandigital Integers
Pandigital Integers remaining Pandigital on Multiplication/Mistake

== Source Work ==
:
:The Dictionary
::$123,456,789$
:
:The Dictionary
::$123,456,789$
== Mistake ==
:There are several numbers that are pandigital, including zero, and remain so when multiplied by several factors. For example, $1,098,765,432$ when multiplied by $2$, $4$, $5$ or $7$.
$8$ has been overlooked:
:$1 \, 098 \, 765 \, 432 \times 8 = 8 \, 790 \, 123 \, 456$
As the article in which this factoid focused on the fact that $123,456,789 \times 8 = 987 \, 654 \, 312$, this omission looks to be a mistake.
== Sources ==
* : $123,456,789$
* : $123,456,789$
Category:Mistakes/Curious and Interesting Numbers
Pandigital Numbers Divisible by All Integers up to 18

== Theorem ==
The following pandigital integers are divisible by all the positive integers up to $18$:
:$2 \, 438 \, 195 \, 760$
:$3 \, 785 \, 942 \, 160$
:$4 \, 753 \, 869 \, 120$
:$4 \, 876 \, 391 \, 520$
== Proof ==
== Sources ==
* : $\text {XI}$. Divisibility: $323$: A Division Curiosity
* : $2,438,195,760$
* : $2,438,195,760$
Category:Pandigital Integers
Category:Pandigital Numbers Divisible by All Integers up to 18
Pandigital Pairs whose Squares are Pandigital

== Theorem ==
The elements of the following pandigital pairs of integers each have squares which are themselves pandigital:
:$\left({35 \, 172, 60 \, 984}\right), \left({57 \, 321, 60 \, 984}\right), \left({58 \, 413, 96 \, 702}\right), \left({59 \, 403, 76 \, 182}\right)$
== Proof ==
== Sources ==
* 
* : $57,321$
Category:Square Numbers
Category:Pandigital Sets
Pandigital Product of Pandigital Pairs in 3 Ways

== Theorem ==
The pandigital integer $0 \, 429 \, 315 \, 678$ can be expressed as the product of a pandigital doubleton in $3$ different ways:
== Proof ==
We have that:
:$0 \, 429 \, 315 \, 678 = 2 \times 3^2 \times 11 \times 19 \times 139 \times 821$
Then:
== Sources ==
* 
* : $0,429,315,678$
* : $0,429,315,678$
Category:Pandigital Sets
Pandigital Properties of 9,876,543,210

== Theorem ==
$9 \, 876 \, 543 \, 210$ has the following properties:
:$9 \, 876 \, 543 \, 210 - 0 \, 123 \, 456 \, 789 = 9 \, 753 \, 086 \, 421$
all three terms of which are pandigital.
== Sources ==
* : $9,876,543,210$
* : $9,876,543,210$
Category:9,876,543,210
Category:Pandigital Integers
Pandigital Sum whose Components are Multiples

== Theorem ==
The following sums use each of the digits $1$ to $9$ exactly once each, while one summand equals $2$ times the other summand:
It is interesting to note how the $4$ sums can be divided into two pairs in which each element of the sum is an anagram of its corresponding element in the other.
== Proof ==
We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital.
Firstly, among the three numbers, there cannot be repeating digits or $0$.
If $5 \divides x$:
:$10 \divides 2 x$ and $2 x$ ends in $0$.
By Divisibility by 5, $x$ cannot end in $5$.
Secondly, $x$ must be divisible by $3$.
To see this, consider the concatenation of the three numbers, $\sqbrk {x || 2 x || 3 x}$.
Since the digits of this number is a permutation of $\set {1, \dots, 9}$, they sum to $45$.
Since $9 \divides 45$, by Divisibility by 9:
:$9 \divides \sqbrk {x || 2 x || 3 x}$
We have:
Hence $9 \divides 6 x$.
This gives $3 \divides 2 x$.
By Euclid's Lemma:
:$3 \divides x$
Finally, as $3 x < 1000$, we must have $100 \le x \le 333$.
Putting repeating digits and zeroes into consideration, we list and check all possible candidates from $123$ to $327$:
Hence the result.
== Sources ==
* 
* : $192$
Category:Recreational Mathematics
Pappus's Hexagon Theorem

== Theorem ==
Let $A, B, C$ be a set of collinear points.
Let $a, b, c$ be another set of collinear points.
Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$.
Then $X, Y, Z$ are collinear points.
:
== Proof ==
== Also known as ==
== Also see ==
* Pascal's Theorem
* Pascal's Mystic Hexagram
* Desargues' Theorem
== Historical Note ==
== Sources ==
* : Chapter $\text{V}$: "Greatness and Misery of Man"
* : Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note
* : $9$
* : Chapter $\text {A}.8$: Pappus (fourth century A.D.)
* : $9$
* 
* 
Category:Pappus's Hexagon Theorem
Category:Pappus's Theorems
Category:Euclidean Geometry
Category:Projective Geometry
Pappus's Hexagon Theorem/Also known as

== Pappus's Hexagon Theorem: Also known as ==
Pappus's Hexagon Theorem is also known just as Pappus's Theorem, but there is more than one such.
Hence on we rigorously use the full version.
== Sources ==
* : Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note
* : $9$
* : Chapter $\text {A}.8$: Pappus (fourth century A.D.)
* : $9$
* 
* 
Category:Pappus's Hexagon Theorem
Pappus's Hexagon Theorem/Historical Note

== Historical Note on Pappus's Hexagon Theorem ==
Pappus's Hexagon Theorem was first proved by in about $300$ CE.
The theorem is stated as Propositions $138$, $139$, $141$, and $143$ of Book $\text{VII}$ of 's .
It is noted that it is a limiting case of Pascal's Mystic Hexagram.
In $1899$ its full significance was revealed by , during his work on clarifying the foundations of geometry.
== Sources ==
* : Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note
* : $9$
* : Chapter $\text {A}.8$: Pappus (fourth century A.D.)
* : $9$
Category:Historical Notes
Pappus's Hexagon Theorem/Proof 1

== Theorem ==
== Proof ==
The notation has been changed to match the source.
Let $ACE$ be collinear and $BFD$ also be collinear.
Join $ABCDEF$ in order.
The points where opposite sides of the hexagon cut each other are $NLM$:
* $AB$ and $DE$ cross at $L$
* $BC$ and $EF$ cross at $N$
* $CD$ and $FA$ cross at $M$
$NLM$ are to be proved collinear.
There are two cases.
Case 1: $EF$ not $\parallel CD$.
Produce $EF$ to meet $CD$ at $U$.
Label points $V$ and $W$ in $\triangle UVW$.
Apply Menelaus's Theorem to five transversals of $\triangle UVW$.
Multiply $(2) \cdot (3) \cdot (4)$:
Multiply $(1) \cdot (5)$:
Multiply $(7)$ by $-1$ and equate the of $(6)$ to the of $(7)$:
By Menelaus's Theorem, $NLM$ are collinear.
Case 2: $EF \parallel CD$ (and $BC \parallel AF$).
$EF \parallel CD$ gives similar triangles.
For the first two, $\angle B$ is shared.
For the next two, $\angle A$ is shared.
Multiply the of $(1)$ through $(5)$ together as the new .
Multiply the of $(1)$ through $(5)$ together as the new .
We have :
:$VLW$ are collinear
Suppose $NLM$ are not collinear.
Then:
:$\angle NLV \ne \angle MLW$
But this is a contradiction, since $\triangle NLV \sim \triangle MLW$ and $\angle NLV = \angle MLW$.
Hence, $NLM$ are collinear.
The result follows.
== Sources ==
* : Chapter $3$ : "Collinearity and Concurrence."
Category:Pappus's Hexagon Theorem
Pappus's Theorem

* Pappus's Hexagon Theorem
* Pappus Centroid Theorems, of which there are $2$ instances:
** Pappus's Centroid Theorem for Volume, also known as the First Pappus-Guldinus Theorem (with )
** Pappus's Centroid Theorem for Surface Area, also known as the Second Pappus-Guldinus Theorem (with )
== Sources ==
* 
Category:Named Theorems/Pappus
Pappus's Theorems

== Theorem ==
There are $2$ theorems which bear the name of :
== Pappus Centroid Theorems, also known as the Pappus-Guldinus Theorems ==
These comprise the $2$ separate theorems:
== Pappus's Hexagon Theorem ==
== Also known as ==
== Sources ==
* 
* 
Category:Pappus's Theorems
Category:Geometry
Pappus-Guldinus Theorems

== Theorem ==
There are two theorems that bear the name Pappus-Guldinus Theorem:
=== First Pappus-Guldinus Theorem ===
=== Second Pappus-Guldinus Theorem ===
== Also known as ==
These theorems are also known as:
:the Pappus Centroid Theorems
:Guldinus' Theorems.
== Also see ==
* Pappus's Theorems
== Sources ==
* : Chapter $\text {A}.8$: Pappus (fourth century A.D.)
* 
Category:Pappus's Theorems
Category:Solids of Revolution
Category:Surfaces of Revolution
Parabolas Inscribed in Shared Tangent Lines

== Theorem ==
Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane.
Let $\map {y_1} x$ be the equation of the tangent line at $\tuple {Q, \map f Q}$ on $f$.
Let $\map {y_2} x$ be the equation of the tangent line at $\tuple {-Q, \map f {-Q} }$ on $f$.
Then there exists another function $\map g x$ also embedded in the Euclidean Plane defined as:
:$\map g x = -A x^2 + B x + C_2$.
with:
:tangent lines $\map {y_3} x$ being the equation of the tangent line at $\tuple {Q, \map g Q}$ on $g$
and:
:$\map {y_4} x$ being the equation of the tangent line at $\tuple {-Q, \map g {-Q} }$ on $g$.
so that the tangent lines $y_3$ and $y_4$ inscribe $\map f x$ and the tangent lines $y_1$ and $y_2$ inscribe $\map g x$.
== Proof ==
The tangent line at $\tuple {Q, \map f \Q}$ on $f$ is defined as:
:$\map {y_1} x = \paren {2 A Q + B} x + b_1$
where $2 A Q + B$ is the slope of the tangent line on the point $\tuple {Q, \map g Q}$ on $f$.
 
Substitute in the coordinates of the point $\tuple {Q, \map g Q}$ to $y_1$ and solve for $b_1$.
This will reveal the $y$-intercept of $y_1$:
 
Continue by following the same steps for $y_2$ which is defined:
:$\map {y_2} x = \paren {-2 A Q + B} x + b_2$
where $-2 A Q + B$ is the slope of the tangent line at the point $\tuple {-Q, \map f {-Q} }$ on $f$.
Substitute in the coordinates of the point $\paren {-Q, \map f {-Q} }$ to $y_2$.
Use these values to solve for $b_2$, and this will reveal the $y$-intercept of $y_2$:
The $y$-intercepts of both $y_1$ and $y_2$ have been shown to be equivalent.
Since $b_1 = b_2$ redefine this value as $b$.
The distance between $b$ and $C_1$ is $\size {C_1 - b}$.
Let $\map g x = -A x^2 + B x + C_2$.
Then the tangent line at the point $\tuple {Q, \map g Q}$ on $g$ is defined as:
:$\map {y_3} x = \paren {-2 A Q + B} x + b_3$
where $-2 A Q + B$ is the slope of the tangent line at $\tuple {Q, \map g Q}$ on $g$.
Solve for $b_3$ using the same methods used for $y_1$ and $y_2$.
This will reveal the $y$-intercept of $y_3$:
:$b_3 = A Q^2 + C_2$
The result also follows for the tangent line $\tuple {-Q, \map g {-Q} }$ on $g$ which is defined:
:$y_4 = \paren {-2 A Q + B} x + b_4$
Solving for $b_4$ yields the result:
:$b_4 = A Q^2 + C_2$
The $y$-intercepts of both $y_3$ and $y_4$ have been shown to be equivalent.
Notice that the derivatives of $f$ and $g$ satisfy:
Then it must be true that:
:$y_1 = y_4$ and $y_2 = y_3$
and the functions $y_1$, $y_2$, $y_3$, and $y_4$ share the same $y$-intercept.
Redefine this the $y$-intercepts of the tangent lines as $b$.
Solve for $C_2$ to determine the vertical translation of $\map g x$:
Therefore the function:
:$\map g x = -A x^2 + B x - \paren {2 A Q^2 + C_1}$
will have tangent lines equivalent to the tangent lines on $\map f x$ at the points $\tuple {Q, \map f Q}$, and $\tuple {-Q, \map f {-Q} }$.
Category:Analytic Geometry
Category:Tangents
Paracompact Countably Compact Space is Compact

== Theorem ==
Let $T = \struct {S, \tau}$ be a countably compact space which is also paracompact.
Then $T$ is compact.
== Proof ==
From the definition of paracompact space, a paracompact space is also a metacompact space.
The result follows from Metacompact Countably Compact Space is Compact.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
Category:Compact Spaces
Category:Countably Compact Spaces
Category:Paracompact Spaces
Paracompact Space is Countably Paracompact

== Theorem ==
Let $T = \struct {S, \tau}$ be a paracompact space.
Then $T$ is a countably paracompact space.
== Proof ==
From the definition, $T$ is paracompact space every open cover of $T$ has an open refinement which is locally finite.
This also applies to all countable open covers.
So every countable open cover of $T$ has an open refinement which is locally finite.
This is precisely the definition for a countably paracompact space.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
Category:Paracompact Spaces
Category:Countably Paracompact Spaces
Paracompact Space is Metacompact

== Theorem ==
Let $T = \struct {S, \tau}$ be a paracompact space.
Then $T$ is also metacompact.
== Proof ==
From the definition, $T$ is paracompact every open cover of $T$ has an open refinement which is locally finite.
Consider some open cover $\UU$ of $T$.
Let $x \in S$.
Then there exists some neighborhood $\exists N_x$ of $x$ which intersects only finitely many elements of $\UU$.
Thus $x$ itself can be in only finitely many elements of $\UU$.
Hence $T$ must be, by definition, metacompact.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness
Category:Paracompact Spaces
Category:Metacompact Spaces
Paracompactness is Preserved under Projections

== Theorem ==
Let $I$ be an indexing set.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.
Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate.
If $\struct {S, \tau}$ is paracompact, then each of $\struct {S_\alpha, \tau_\alpha}$ is also paracompact.
== Proof ==
We are given that $\struct {S, \tau}$ is paracompact.
So every open cover of $S$ has an open refinement which is locally finite.
We are given that for each $\alpha \in I$, $\pr_{\alpha}: S \to S_{\alpha}$ is the $\alpha$-th projection.
For each $\alpha \in I$, let $\pr_{\alpha}^\gets: \powerset {S_{\alpha}} \to \powerset S$ be the inverse image mapping induced by $\pr_{\alpha}$
Let $T \subseteq S_\alpha$.
Let $\family {T_i}_{i \mathop \in I}$ be the family of sets defined by:
:$T_i = \begin {cases} T & : i = \alpha \\ S_i & : i \ne \alpha \end {cases}$
By Inverse Image Mapping Induced by Projection, for all $\alpha \in I$, $\pr_\alpha^\gets$ is the mapping defined by:
:$\ds \forall T \subseteq S_i: \map {\pr_{\alpha}^\gets} T = \prod_{i \mathop \in I} T_i$
Therefore every open cover of $\struct {S_\alpha, \tau_\alpha}$ has an open refinement which is locally finite.
Hence every $\struct {S_\alpha, \tau_\alpha}$ is paracompact.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties
Category:Paracompact Spaces
Category:Continuous Mappings
Category:Product Spaces
Category:Projections
Paracompactness is not always Preserved under Open Continuous Mapping

== Theorem ==
Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact.
Let $T_B = \struct {X_B, \tau_B}$ be another topological space.
Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open.
Then it is not necessarily the case that $T_B$ is also paracompact.
== Proof ==
We have Open Continuous Image of Paracompact Space is not always Countably Metacompact.
We also have:
:Paracompact Space is Metacompact
:Metacompact Space is Countably Metacompact
Hence the result.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties
Category:Paracompact Spaces
Category:Continuous Mappings
Category:Product Spaces
Category:Open Mappings
Paradoxes of Material Implication

== Theorems ==
The conditional operator has the following counter-intuitive properties:
=== Tautological Consequent ===
=== Tautological Antecedent ===
=== Contradictory Antecedent ===
=== Contradictory Consequent ===
== Also presented as ==
These results are also presented in the following forms:
=== True Statement is implied by Every Statement ===
If something is true, then anything implies it.
=== False Statement implies Every Statement ===
If something is false, then it implies anything.
Also note this counterintuitive result:
=== Disjunction of Conditional and Converse ===
== Examples ==
== Historical Note ==
== Sources ==
* : $\S \text{II}.13$: Symbolism of sentential calculus
* : $\S 2.3$: Basic Truth-Tables of the Propositional Calculus
* : Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules
* : $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms
* : $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Conditional
* 
* 
Category:Paradoxes of Material Implication
Category:Conditional
Category:Veridical Paradoxes
Paradoxes of Material Implication/Examples/Red Grass and Green Moon

== Example of Paradox of Material Implication ==
The compound statement:
:If grass is red then the moon is made of green cheese
is true, despite being semantically meaningless.
== Sources ==
* : $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Conditional
Category:Examples of Paradoxes of Material Implication
Paradoxes of Material Implication/Historical Note

== Historical Note on Paradoxes of Material Implication ==
The Paradoxes of Material Implication have caused debate and puzzlement among philosophers for millennia.
In particular, the result $\neg p \vdash p \implies q$ is known as a vacuous truth. It is exemplified by the (rhetorical) argument:
:If England win the Ashes this year, then I'm a monkey's uncle.
Category:Historical Notes
Paradoxes of Strict Implication

== Theorems ==
The strict implication operator has the following counter-intuitive properties:
=== Necessary Proposition is Strictly Implied by Every Proposition ===
If a proposition is necessarily true, then every proposition strictly implies it.
=== Impossible Proposition Strictly Implies Every Proposition ===
If proposition is not possibly true, then it strictly implies every proposition.
== Examples ==
== Sources ==
* 
* 
Category:Paradoxes of Strict Implication
Category:Strict Implication
Category:Veridical Paradoxes
Parallactic Angle in terms of Azimuth and Declination

== Theorem ==
Let $x$ be a point on the celestial sphere.
Let $q$ denote the parallactic angle of $x$.
Then:
:$\sin q := \dfrac {\cos \phi \sin A} {\sin \delta}$
where:
:$\phi$ is the terrestrial latitude
:$A$ is the azimuth
:$\delta$ is the declination.
== Proof ==
== Sources ==
* 
* 
Category:Parallactic Angles
Category:Terrestrial Latitude
Category:Azimuth (Astronomy)
Category:Declination
Parallactic Angle in terms of Hour Angle and Zenith Distance

== Theorem ==
Let $x$ be a point on the celestial sphere.
Let $q$ denote the parallactic angle of $x$.
Then:
:$\sin q := \dfrac {\cos \phi \sin t} {\sin \zeta}$
where:
:$\phi$ is the terrestrial latitude
:$t$ is the hour angle
:$\zeta$ is the zenith distance.
== Proof ==
== Sources ==
* 
* 
Category:Parallactic Angles
Category:Terrestrial Latitude
Category:Hour Angle
Category:Zenith Distance
Parallel Chords Cut Equal Chords in a Circle

== Theorem ==
Let $\CC$ be a circle on center $O$.
Let $AB$ and $CD$ be chords in $\CC$ with $AB \parallel CD$.
Then the two chords cut from the circle by $AB$ and $CD$ are equal:
:$AC = BD$
== Proof ==
Draw $AD$.
By Parallelism implies Equal Alternate Angles:
:$\angle BAD = \angle ADC$
The central angles corresponding to $\angle BAD$ and $\angle ADC$ are equal by Inscribed Angle Theorem:
:$\angle BOD = \angle AOC$
As radii of $\CC$:
:$OB = OD = OC = OA$
By Triangle Side-Angle-Side Congruence:
:$\triangle OAC \cong \triangle OBD$
So:
:$AC = BD$
Category:Circles
Category:Parallel Lines
Parallel Elements Depend on Same Subsets

== Theorem ==
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $x, y \in S$.
Let $x$ be parallel to $y$.
Then:
:$x$ depends on $A$ $y$ depends on $A$
== Proof ==
This follows directly from Closure of Subset Contains Parallel Elements.
== Sources ==
* Chapter $1.$ $\S 4.$ Loops and parallel elements
Category:Matroid Parallelism
Category:Matroid Dependence