Problem
GEO-B3-M06-P011 Isogonal Conjugation and Ceva
#11
★★★☆☆ Level 3 of 5
Let cevians \(AA_1,BB_1,CC_1\) be concurrent. Prove that their isogonal cevians are also concurrent.
Inspired by Prasolov trigonometric geometry method
C. Hint 1. Write trig Ceva for the original cevians.
D. Hint 2. Under isogonal reflection, each factor is inverted.
For the original cevians, trig Ceva gives \[ X_A X_B X_C=1, \] where \(X_A=\frac{\sin\angle BAA_1}{\sin\angle CAA_1}\), and similarly.
The isogonal cevian at angle \(A\) swaps the two angles with \(AB\) and \(AC\), so the new factor is \(\frac1{X_A}\). Similarly, the others are \(\frac1{X_B}\), \(\frac1{X_C}\). Their product is again \(1\). By trig Ceva, the isogonal cevians are concurrent.
A short but important proof of the existence of isogonal conjugation.