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 4 of 5: Apply the two midline theorems
Detailed analysis
Because M and Q are midpoints of BC and BR, MQ is a midline in triangle BCR. The analogous argument with S, the midpoint of AB, shows that P is the midpoint of CS and MP is a midline in triangle BCS.