SELECTION and TRAINING SESSION

Question

An arbitrary point D is marked inside the triangle ABC. The lines AD, BD and CD intersect the sides BC, CA and AB at the points K, L and M respectively. Let A_1, B_1 and C_1 be the midpoints of the segments BC, CA and AB, and X, Y be the midpoints of the segments ML and KM respectively. The lines XA_1 and YB_1 intersect at the point F. Prove that FO DH, where O is the circumcenter of the triangle A_1B_1C_1 and H is the orthocenter of the triangle ABC. (Mikhail Karpuk)

Accepted Answer

Note that XA_1 is the Gauss-Newton line of a complete quadrangle formed by the lines BL, BA and CM, CA, hence it passes through the midpoint of the segment AD. Similarly YB_1 passes through the midpoint of the segment BD. Therefore F is the center of mass of points A, B, C and D with unit masses. This implies that F lies on the line DG and GF: FD = 1:3 where G is the centroid of the triangle ABC. At the same time O is the nine-point center of the triangle ABC so it also lie on the Euler line of the triangle ABC and GO: OH = 1:3. Therefore the Thales's theorem implies that FO DH.