Problem
GEO-B3-M05-P006 Tangents Give a Symmedian
The tangents to the circumcircle of \(ABC\) at \(B\) and \(C\) meet at \(P\). Prove that \(AP\) contains the \(A\)-symmedian of the triangle.
C. Hint 1. Let \(AP\cap BC=S\).
D. Hint 2. Prove \(\frac{BS}{CS}=\frac{AB^2}{AC^2}\) using tangent angles.
By the tangent-chord theorem, \(\angle PBA=\angle ACB\), \(\angle PCB=\angle CAB\), with analogous equalities for the other tangent. Let \(S=AP\cap BC\).
Applying the sine rule in triangles \(PBS\) and \(PCS\), we get \(\frac{BS}{CS}=\frac{PB\sin\angle BPS}{PC\sin\angle SPC}\). Since \(PB=PC\), and the angles at \(P\) are expressed through arcs \(AB\) and \(AC\), the sine rule in \(ABC\) gives \(\frac{BS}{CS}=\frac{AB^2}{AC^2}\).
By the symmedian criterion, \(AS\), hence \(AP\), is the \(A\)-symmedian.
This problem is a common bridge from circles to Lemoine geometry.