# Symmedian in a Right Triangle

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet suggests a simple fact that, in a right triangle, the symmedian to the hypotenuse coincides with the altitude from the right angle.

A *symmedian* is the isogonal conjugate of a median from the same vertex. Thus, for example, in right triangle ABC, with the right angle at C, if CM is the median and CH is the symmedian through C, then angles ACM and BCH are equal. But in a right triangle, the median through the right angle equals half the hypotenuse, so that triangle AMC is isosceles. Its base angles MCA and MAC are equal. We thus have

∠CAB = ∠BCH,

∠CAB + ∠ABC = 90°, and

∠ABC = ∠HBC.

Therefore

∠HBC + ∠CBH = 90°.

It thus follows that angle CHB is right, as asserted.

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Copyright © 1996-2018 Alexander Bogomolny