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Bundeswettbewerb Mathematik 1993 Problem 2.3

Source: Bundeswettbewerb Mathematik 1993 Round 2

October 10, 2022
geometryperpendicular bisectorangle bisectorTriangle

Problem Statement

In the triangle ABCABC, let A′A' be the intersection of the perpendicular bisector of ABAB and the angle bisector of ∠BAC\angle BAC and define B′,C′B', C' analogously. Prove that
a) The triangle ABCABC is equilateral if and only if A′=B′.A' =B'.
b) If A′,B′A', B' and C′C' are distinct, we have ∠B′A′C′=90∘−12∠BAC.\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.
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