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Parallelogram and congruent triangles

Source: Irish Mathematical Olympiad 2023 Problem 1

May 15, 2023

geometryparallelogramcongruent triangles

Problem Statement

We are given a triangle

ABC

such that

\angle BAC < 90^{\circ}

. The point

D

is on the opposite side of the line

AB

to

C

such that

|AD| = |BD|

and

\angle ADB = 90^{\circ}

. Similarly, the point

E

is on the opposite side of

AC

to

B

such that

|AE| = |CE|

and

\angle AEC = 90^{\circ}

. The point

X

is such that

ADXE

is a parallelogram.
Prove that

|BX| = |CX|

.

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