Prove Geometric Theorem

Solve trigonometric equations and triangle problems systematically using identities, law of sines/cosines, inverse functions, and unit circle analysis. Covers equation solving, triangle resolution, identity verification, and applied trigonometric modeling. Use when solving trigonometric equations for unknown angles, resolving triangles from partial information (SSS, SAS, ASA), verifying identities, or applying trigonometry to real-world problems in surveying, physics, or engineering.

Derive a theoretical result step-by-step from first principles or established theorems, with every step explicitly justified and special cases checked. Use when deriving a formula or theorem from first principles, proving a mathematical statement by logical deduction, re-deriving a textbook result for verification or adaptation, extending a known result to a more general setting, or producing a self-contained derivation for a paper or thesis.

Analyze prime numbers using primality tests, factorization algorithms, prime distribution analysis, and sieve methods. Covers trial division, Miller-Rabin, Sieve of Eratosthenes, and the Prime Number Theorem. Use when determining whether an integer is prime or composite, finding prime factorizations, counting or listing primes up to a bound, or investigating prime properties within a number-theoretic proof or computation.

Construct well-structured arguments using the hypothesis-argument-example triad. Covers formulating falsifiable hypotheses, building logical arguments (deductive, inductive, analogical, evidential), providing concrete examples, and steelmanning counterarguments. Use when writing or reviewing PR descriptions that propose technical changes, justifying design decisions in ADRs, constructing substantive code review feedback, or building a research argument or technical proposal.

Solve Diophantine equations (integer-only solutions) including linear, quadratic, and Pell equations. Covers the extended Euclidean algorithm, descent methods, and existence proofs. Use when finding all integer solutions to ax + by = c, solving Pell's equation, generating Pythagorean triples, proving no integer solutions exist via modular constraints, or finding the fundamental solution from which all others are generated.

Perform a ruler-and-compass construction with step-by-step justification for each operation, producing a constructible geometric figure from given elements. Covers classical Euclidean constructions including perpendicular bisectors, angle bisectors, parallel lines, regular polygons, and tangent lines. Use when given geometric elements (points, segments, angles) and asked to produce a Euclidean construction, verify constructibility, or generate ordered construction steps for educational or documentation purposes.