Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.6 AIME, harder Prove it

132. The line passing through the midpoints of the diagonals $A C$ and $B D$ of quadrilateral $A B C D$ intersects sides $A B$ and $D C$ at points $M$ and $N$ . Prove that $S_{D C M}=S_{A N B}$ .

Solution

132. Let $K$ be the midpoint of $DB$ , and $L$ be the midpoint of $AC$ . $S_{ANM}=S_{CNM}$ (since $|AL|=|LC|$ ), similarly $S_{BNM}=S_{DMN}$ , from which the statement of the problem follows.

Fig. 34.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.