MathLabs

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 5 of 5: Conclude the similarity
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MP/MQ=AB/ACand both included angles equal ∠BAC⟹△MPQ∼△ABC.MP/MQ=AB/AC\quad\text{and both included angles equal }\angle BAC\Longrightarrow\triangle MPQ\sim\triangle ABC.
Detailed analysis

The two sides from M are parallel to AB and AC and have the same factor 1/4. Thus the included angle and the corresponding side ratio agree, proving that MPQ is similar to ABC.