Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it United States

Problem:

Two congruent line segments $AB$ and $CD$ intersect at a point $E$ . The perpendicular bisectors of $AC$ and $BD$ intersect at a point $F$ in the interior of $\angle AEC$ . Prove that $EF$ bisects $\angle AEC$ .

Solution

Solution:

Since $F$ is on the perpendicular bisectors of $AC$ and $BD$ , $FA = FC$ and $FB = FD$ . Also $AB = CD$ is given. Thus $\triangle FAB \cong \triangle FCD$ by SSS. Since corresponding heights of congruent triangles are equal, $F$ is equidistant from $AB$ and $CD$ , implying that $F$ is on the bisector of $\angle AEC$ .

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