Problem
GEO-B3-M05-P005 Existence of the Lemoine Point
#5
★★☆☆☆ Level 2 of 5
Prove that the three symmedians of triangle \(ABC\) are concurrent.
Inspired by Prasolov special points geometry method
C. Hint 1. Use the symmedian criterion.
D. Hint 2. Multiply the three ratios on the sides and apply Ceva's theorem.
Let the \(A\)-symmedian meet \(BC\) at \(A_1\). Then \(\frac{BA_1}{CA_1}=\frac{AB^2}{AC^2}=\frac{c^2}{b^2}\). Similarly, for \(B_1\in CA\), \(C_1\in AB\), we have \(\frac{CB_1}{AB_1}=\frac{a^2}{c^2}\), \(\frac{AC_1}{BC_1}=\frac{b^2}{a^2}\).
The product is \(1\). By Ceva's theorem, \(AA_1,BB_1,CC_1\) are concurrent. This point is the Lemoine point \(K\).
For extensions, oriented ratios should be used.