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Problem 1097

AMC 12 late, AIME early

Geometry Difficulty 4.7 Prove it Ukrainian National Mathematical Olympiad · Ukraine

Cut right triangle with one angle $30^\circ$ into three pairwise distinct isosceles triangles so that each has a non acute angle.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

We can cut $\triangle ABC$ with $\angle BAC = 30^\circ$ into triangles as shown in Fig. 25.

Here $BC = CD$ and $\triangle DBC$ is isosceles with right angle. Then $BN = MD$ and $\triangle DBM$ is isosceles with angle $\angle DMB = 150^\circ$ . Thus $\angle DMA = 30^\circ$ , $\triangle ADM$ is isosceles with $\angle ADM = 120^\circ$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.