Competition problem — Geometry — Olympiad Maths Prep

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Geometry Difficulty 4.7 AIME Prove it Ukraine

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

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 $\angle ADM = 120^\circ$ .

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