Brute-forcing Langley’s geometry problem with field extensions
a day ago
- Langley's Adventitious Angles is a classic geometry problem to find an angle in a specific diagram, often solved by a clever construction.
- The problem is difficult because standard angle chasing leads to equations where the unknown angle cancels out, suggesting non-uniqueness.
- The angle is uniquely determined by the geometry, but approximate methods (drawing, trigonometry) only give an approximate answer.
- The article presents a systematic method using complex numbers and polynomial algebra to compute the exact answer without clever insights.
- The method represents points as polynomials in t = e^(2πi/36) (a 10° rotation) and performs arithmetic modulo the minimal polynomial t^12 - t^6 + 1.
- All operations (addition, multiplication, division, conjugation) are done exactly using polynomial arithmetic, yielding an exact result for the final angle.
- For the original 80°-80°-20° triangle, the computed angle is exactly 30°, as verified by the polynomial calculation returning q = t^6.
- The method can handle variants (e.g., different initial angles) and still gives exact algebraic expressions, though the final angle may be irrational.
- The approach avoids trigonometry and is based on field extensions and Galois theory, making it a reusable, systematic technique.