MathDB

P1

Part of 2023 Irish Math Olympiad

Problems(1)

Parallelogram and congruent triangles

Source: Irish Mathematical Olympiad 2023 Problem 1

5/15/2023
We are given a triangle ABCABC such that ∠BAC<90∘\angle BAC < 90^{\circ}. The point DD is on the opposite side of the line ABAB to CC such that ∣AD∣=∣BD∣|AD| = |BD| and ∠ADB=90∘\angle ADB = 90^{\circ}. Similarly, the point EE is on the opposite side of ACAC to BB such that ∣AE∣=∣CE∣|AE| = |CE| and ∠AEC=90∘\angle AEC = 90^{\circ}. The point XX is such that ADXEADXE is a parallelogram.
Prove that ∣BX∣=∣CX∣|BX| = |CX|.
geometryparallelogramcongruent triangles