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Innovative geo with areas and angles 18 degrees

Source: ARO Regional stage 2024 11.4

February 2, 2024
geometry

Problem Statement

Let XYXY be a segment, which is a diameter of a semi-circle. Let ZZ be a point on XYXY and 9 rays from ZZ are drawn that divide ∠XZY=180∘\angle XZY=180^{\circ} into 1010 equal angles. These rays meet the semi-circle at A1,A2,…,A9A_1, A_2, \ldots, A_9 in this order in the direction from XX to YY. Prove that the sum of the areas of triangles ZA2A3ZA_2A_3 and ZA7A8ZA_7A_8 equals the area of the quadrilateral A2A3A7A8A_2A_3A_7A_8.
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