# A Characteristic Property Of Right Triangles

Let in ΔABC, CM is a median from C, CH is an altitude.

I came across this property in an article *Characterizations of Bicentric Quadrilaterals* by Martin Josefsson (Forum Geometricorum, Volume 10 (2010) 165-173.) The proof below is different from that in the article.

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Copyright © 1996-2017 Alexander Bogomolny

Let in ΔABC, CM is a median from C, CH is an altitude.

### Proof

First assume that ∠ACB = 90°. Then AB is a diameter of the circumcircle. In particular,

If S is the reflection of C in H then also ΔBCS is isosceles so that

Conversely, assume ∠ACM = ∠BCH. Draw the angle bisector of angle ACB:

The bisector intersects the circumcircle in L - the midpoint of arc AB - such that LM ⊥ AB. Since

### Corollary

In the previous notations, ∠ACB is right iff its bisector also bisects ∠MCH.

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Copyright © 1996-2017 Alexander Bogomolny

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