Research task (STEM):
Write a blog post contrasting Euclidean geometry used in high school math competitions such as the AMC, AIME, or Olympiads to geometry taught in the standard American high school math curriculums. This post is written for people with knowledge of geometry from standard high school classes, but with no math experience in competitive math. Your post must include clear examples of problem solving techniques and theorems that are commonly used in high school math competitions, but not typically taught in a high school math class.

Evaluation rubric:
criterion: The response ensures all acronyms are expanded on it during their first appearance and provides clear definitions for all specialized terms and mathematical symbols upon first use, accompanied by explanatory prose or examples (e.g., (1) SSS, meaning, Side-Side-Side, criteria to establish that two triangles are congruent, (2) American Invitational Mathematics Examination (AIME), (3) American Mathematics Competitions (AMC), (4) United States of America Mathematical Olympiad (USAMO)).
weight: 4
axis: Communication Quality

criterion: The response includes a geometric diagram for each theorem and sample problem introduced (e.g., (1) Draw an example of a cyclic quadrilateral and the relevant angles that sums to 180, (2) Draw and example cyclic quadrilateral with the diagonals to show the Ptolemy's Theorem, (3) Draw an example of a triangle of the cevians to illustrate the Ceva's Theorem, (4) Draw three circles to illustrate the Collinearity of External Tangent Intersections).
weight: 4
axis: Implicit Criteria

criterion: The response includes a geometric diagram far from the text where it is discussed (i.e., the text describes an image that is missing or is on pages away).
weight: -3
axis: Communication Quality

criterion: The response keeps the primary focus of the blog post on Euclidean geometry (e.g. (1) Ceva's Theorem, (2) Menelaus' Theorem, (3) Power of a Point Theorem, (4) Stewart's Theorem, (5) Ptolemy's Theorem, (6) Spiral Similarity, (7) Miquel Point) and does not include problems from other topics such as functions and combinatorics.
weight: 5
axis: Instruction Following

criterion: The response's blog post contains at least 2 sub-headings to separate discussion of different topics (e.g., Subheadings titled: (1) Advanced Theorems, (2) Advanced Problem-Solving Techniques, (3) Example Problems and Solutions, (4) Where to go from here?).
weight: 2
axis: Communication Quality

criterion: The response's blog post mentions at least one high school math competition (e.g., (1) AMC, (2) AIME, (3) USA(J)MO, or (4) IMO).
weight: 3
axis: Instruction Following

criterion: The response's blog post elaborates on a theorem in the common core high school geometry standards, beyond a quick definition to contrast it with competition-level requirements (e.g., Pythagoras' Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides. This can be used to find the missing lengths of the right triangle as long as two other lengths are known, and continues to provide a diagram and example usage).
weight: 2
axis: Implicit Criteria

criterion: The response identifies at least one theorem in competition-style Euclidean geometry problems (e.g., (1) Ceva's Theorem, (2) Menelaus' Theorem, (3) Power of a Point Theorem, (4) Stewart's Theorem, (5) Ptolemy's Theorem, (6) Spiral Similarity, (7) Miquel Point).
weight: 5
axis: Explicit Criteria

criterion: The response identifies at least one problem-solving technique in Euclidean geometry (e.g., (1) Correctly identifying what theorems to use, (2) Correctly applying the theorem, (3) Adding auxiliary lines, (4) Homothety, (5) Symmetries, rotations, and reflections, (6) Inversion).
weight: 5
axis: Explicit Criteria

criterion: The response introduces a problem-solving technique that is included in the common core high school geometry standards (e.g., (1) Pythagoras' Theorem, (2) Midpoint Theorem, (3) Congruent Triangle Theorems, (4) Angle Bisector Theorem) and provides a sample problem that can be solved using the technique.
weight: -4
axis: Implicit Criteria

criterion: The response includes a sample problem to demonstrate problem-solving techniques (e.g., (1) AMC 12 2018 #8, (2) 2020 AIME II Problem 15, (3) 2020 AMC 12A Problem 18, (4) 2021 USAJMO Problem 3, or (5) explicitly stating a simple instance of a problem where the techniques can be applied).
weight: 3
axis: Implicit Criteria

criterion: The response describes the test format of at least one high school math competition (e.g., (1) The AIME is a 15-question, 3-hour exam, (2) Each answer is an integer from 000 to 999, inclusive, (3) The AMC 10 is a 25-question, 75-minute multiple-choice test. (4) Calculators are not permitted).
weight: 2
axis: Implicit Criteria

criterion: The response incorrectly defines a theorem, incorrectly presents the solution to a sample problem, or incorrectly provides information about a high school math competition.
weight: -5
axis: Implicit Criteria

criterion: The response identifies a theorem or problem-solving technique that is discussed within the first 200 words of the post (e.g., Ptolemy's theorem establishes the relationship between the length of the sides (abcd) and the diagonals (pq) of a cyclic quadrilateral where ac + bd = pq).
weight: 2
axis: Communication Quality

criterion: The response's blog post presents a common core style math problem as an olympiad style math problem (e.g., (1) Pythagoras' Theorem, (2) Midpoint Theorem, (3) Congruent Triangle Theorems, (4) Angle Bisector Theorem).
weight: -3
axis: Implicit Criteria

criterion: The response identifies a distinction between competition-style math problems and standard curriculum math problems (e.g., (1) Competition-style math problems cover more breadth and depth using more advanced theorems and problem-solving skills, (2) Competition-style math problems require more "tricks", (3) Competition-style math problems require more steps of problem-solving involving unique combinations of applications of math theorems, (4) Competition-style math problems are highly specialized to specific topics such as geometry, combinatorics, and number theory, and less on analysis and algebra.
weight: 4
axis: Synthesis of Information

criterion: The response includes a step-by-step solution to a sample problem introduced (e.g., $CAROLINE$ is essentially a plus sign with side length 1 with a few diagonals, which motivates us to coordinate bash. We let $N = (1, 0)$ and $E = (0, 1)$. To find $CORNELIA$'s self-intersections, we take
\[CO = y = 2, AI = y = -3x + 6, RN = y = 3x - 3\]
And plug them in to get $C_1 = \left(\frac{4}{3}, 2 \right)$ where $C_1$ is the intersection of $CO$ and $AI$, and $C_2 = \left(\frac{5}{3}, 2 \right)$ is the intersection of $RN$ and $CO$.
We also track the intersection of $AI$ and $RN$ to get $\left(\frac{3}{2}, \frac{3}{2} \right)$.
By vertical symmetry, the other 2 points of intersection should have the same x-coordinates. We can then proceed with Solution 1 to calculate the area of the triangle (compare the $y$-coordinates of $A,R,I,N$ and $CO$ and $EL$).).
weight: 4
axis: Synthesis of Information

criterion: The response contains messy, unnecessary formulas, code, or tables. (e.g., (1) Unrendered latex, (2) messy formulas with improperly formatted structure or messy comments, (3) broken tables that are unrendered, (4) missing figures or formulas or sections that are referenced in the text).
weight: -4
axis: Communication Quality

criterion: The response cites the source of a sample problem incorrectly (i.e., quotes or explains a particular problem, such as AMC 12 2018 #4, but is actually about a different problem or a variant of the problem that is different).
weight: -3
axis: References & Citation Quality

criterion: The response uses and cites the source of competition problems (e.g., (1) AMC 12 2018 #8, (2) 2020 AIME II Problem 15, (3) 2020 AMC 12A Problem 18, (4) 2021 USAJMO Problem 3).
weight: 2
axis: References & Citation Quality