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 3 of 5: Find Q on the median

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\triangle QGX\sim\triangle QBC,\quad QG/QB=GX/BC=1/3\quad\Longrightarrow\quad BQ=BR/2.

Detailed analysis

Let R be the midpoint of AC. Since G lies on BR and BG=2BR/3, the similarity of QGX and QBC gives QG/QB=1/3. Combining this with BG=QB-QG yields BQ=BR/2, so Q is the midpoint of BR.