Problem 1
Let G be the centroid of triangle ABC and M the midpoint of BC. Points X on AB and Y on AC are such that X, Y, G are collinear and XY is parallel to BC. Suppose XC and GB meet at Q, while YB and GC meet at P. Prove that triangle MPQ is similar to triangle ABC.
Step 2 of 5: Locate G on the parallel section
Detailed analysis
The homothety centered at A sends B,C to X,Y and sends M to the midpoint of XY. Its image of the centroid is therefore that midpoint, so G is the midpoint of XY and GX=BC/3.