Problem
GEO-B3-M05-P008 Symmedian and an Antiparallel
In triangle \(ABC\), segment \(B_1C_1\) with endpoints on rays \(AC\) and \(AB\) is antiparallel to side \(BC\). Prove that the \(A\)-symmedian passes through the midpoint of \(B_1C_1\).
C. Hint 1. Antiparallel means two pairs of equal angles.
D. Hint 2. Compare the triangles near \(A\) and apply the symmedian criterion.
Let \(L\) be the midpoint of \(B_1C_1\), and let \(AL\) meet \(BC\) at \(S\). From antiparallelism, \(\triangle AB_1C_1\sim \triangle ACB\) with opposite orientation. Hence the distance ratios from \(L\) to \(AB\) and \(AC\) are expressed through \(AB\) and \(AC\).
Using areas of triangles \(ABS\) and \(ACS\), we get \(\frac{BS}{CS}=\frac{AB^2}{AC^2}\). By the symmedian criterion, \(AS\) is the \(A\)-symmedian. Therefore \(A,L,S\) are collinear, and the symmedian passes through the midpoint of the antiparallel.
Useful when a symmedian is hidden behind an antiparallel segment.