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 1 of 5: Use the centroid on the median
Detailed analysis
The centroid divides every median in the ratio 2:1 from the vertex. Since XY is parallel to BC, triangles AXY and ABC are similar, so their scale factor is AG/AM=2/3.