Question 126
Points $A_1, B_1, C_1$ are the midpoints of the sides $BC$, $AC$, $AB$ of the triangle $ABC$ ($AB \ge AC \ge BC$), respectively. Points $C_2$ and $C_3$ are marked on $AB$ so that the perimeter of the triangle $BA_1C_2$ is equal to the perimeter of the quadrilateral $ACA_1C_2$ and the perimeter of the triangle $AB_1C_3$ is equal to the perimeter of the quadrilateral $BCB_1C_3$. Point $B_2$ is marked on $AC$ so that the perimeter of the triangle $AB_2C_1$ is equal to the perimeter of the quadrilateral $BCB_2C_1$.
Prove that the segments $A_1C_2$, $B_1C_3$, $C_1B_2$ are concurrent.
(V. Karamzin)
Prove that the segments $A_1C_2$, $B_1C_3$, $C_1B_2$ are concurrent.
(V. Karamzin)