Problem
GEO-B3-M06-P014 Kiepert Lines
On sides \(BC,CA,AB\), external similar isosceles triangles with common outer apex angle \(\varphi\) are constructed. Prove that the lines from \(A,B,C\) to the corresponding outer vertices are concurrent.
C. Hint 1. Find the pair of angles made by the first such line with the sides of angle \(A\).
D. Hint 2. Trig Ceva will give a telescoping product.
Let the outer vertex on side \(BC\) be \(A_1\). From similarity of the constructed isosceles triangles, the direction \(AA_1\) makes angles at \(A\) whose sine ratio is \(\frac{\sin(B+\psi)}{\sin(C+\psi)}\), where \(\psi\) depends only on \(\varphi\).
Cyclically, the three factors are \[ \frac{\sin(B+\psi)}{\sin(C+\psi)} \frac{\sin(C+\psi)}{\sin(A+\psi)} \frac{\sin(A+\psi)}{\sin(B+\psi)}=1. \] By trig Ceva, the lines are concurrent.
This prepares Kiepert points and cubics; here only trigonometric concurrence is needed.