Tilt

A brainteaser by Y. Bogaturov, via Quantum, November-December 1991:

In square ABCD, point L divides diagonal AC in the ratio 3:1 and K is the midpoint of side AB. Prove that angle KLD is a right angle.

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If we divide the square into 16 smaller squares, as shown, K and L fall at nodes in the resulting grid. Now rotating the diagram through 90° around L carries triangle LMD into triangle LNK. So angle DLK must equal the angle of rotation, or 90°.