Competition problem — Geometry — Olympiad Maths Prep

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Geometry Difficulty 4.9 AIME Prove it Ukraine

On the sides $AB$ and $BC$ of the square $ABCD$ we marked points $K$ and $L$ correspondingly so that $KB = LC$ . Segments $AL$ and $CK$ intersect at point $P$ . Prove that lines $DP$ and $KL$ are perpendicular.

Solution

By construction $\Delta ADK = \Delta BAL$ . Then $\angle BAL + \angle ALB = \angle BAL + \angle AKD = 90^\circ$ , and $\Delta AMK$ has a right angle $M$ , so $AL \perp DK$ (fig. 14).

Similarly $DL \perp CK$ . Then lines $AL$ and $CK$ , which intersect at point $P$ , contain two altitudes of the $\Delta DKL$ , and therefore the line $DP$ has to contain the third altitude, so $KL \perp DP$ , as desired.

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