Problem
GEO-B3-M06-P008 Isogonal Pair on a Side
#8
★★☆☆☆ Level 2 of 5
Lines \(AX\) and \(AY\) are isogonal in angle \(A\) of triangle \(ABC\) and meet \(BC\) at \(X,Y\). Prove that \(\frac{BX}{CX}\cdot\frac{BY}{CY}=\frac{AB^2}{AC^2}\).
Inspired by Prasolov trigonometric geometry method
C. Hint 1. Write the ratio formula for each cevian.
D. Hint 2. Isogonality swaps the sine numerator and denominator.
By the cevian formula, \[ \frac{BX}{CX}=\frac{AB\sin\angle BAX}{AC\sin\angle XAC},\quad \frac{BY}{CY}=\frac{AB\sin\angle BAY}{AC\sin\angle YAC}. \] Isogonality gives \(\angle BAX=\angle YAC\), \(\angle XAC=\angle BAY\).
Multiplying cancels the sine factors and leaves \(\frac{AB^2}{AC^2}\).
This is a key to symmedians and isogonal conjugation.