Problem
GEO-B1-M04-P021 The Segment Between Diagonals on the Midline
In trapezoid \(ABCD\), bases \(AD\parallel BC\), \(AD>BC\). Points \(M\) and \(N\) are the midpoints of legs \(AB\) and \(CD\). Diagonals \(AC\) and \(BD\) meet the midline \(MN\) at points \(P\) and \(Q\). Prove that \(PQ=\frac{AD-BC}{2}\).
Prove that \(P\) and \(Q\) are the midpoints of diagonals \(AC\) and \(BD\). Then find the two segments from \(M\) to these points.
Since \(M\) is the midpoint of \(AB\) and \(MN\parallel BC\), in triangle \(ABC\), point \(P\) is the midpoint of \(AC\), and \(MP=\frac{1}{2}BC\). Similarly, in triangle \(ABD\), the line through midpoint \(M\), parallel to \(AD\), meets \(BD\) at its midpoint \(Q\), and \(MQ=\frac{1}{2}AD\). Since \(AD>BC\), point \(Q\) lies farther from \(M\) than \(P\), so \(PQ=MQ-MP=\frac{AD}{2}-\frac{BC}{2}=\frac{AD-BC}{2}\).
A strong midline problem: the student must see two different midlines in two triangles.