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## Where can the perpendicular bisectors of the sides of a right triangle intersect?

Let ACE be a right triangle with C being the vertex at the right angle. Let B be the midpoint of AC and let D be the midpoint of CE.

Draw the perpendicular bisector of AC: by definition, it meets AC at B and at a right angle. Call the point where it intersects the hypotenuse F.

Triangle ABF shares angle A with the original triangle ACE, and also has a right angle, so ABF and ACE are similar triangles. Because AB is 1/2 of AC, AF is 1/2 of AE. F is the midpoint of AE.

Likewise, the perpendicular bisector of CE also intersects AE at midpoint F.

Since two nonparallel lines can only intersect at one point, the two perpendicular bisectors intersect only at F.

A

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C D E

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