Symmedian in a Right Triangle: What is this about?
A Mathematical Droodle

Explanation

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 = 90o, and ABC = HBC.

Therefore

HBC + CBH = 90o.

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

The Symmedians

1. Symmedian and 2 Antiparallels
2. Symmedian and Antiparallel
3. Symmedian in a Right Triangle
4. Nobbs' Points and Gergonne Line
5. Three Tangents Theorem
6. A Tangent in Concurrency
7. Symmedian and the Tangents
8. Bride's Chair
9. Star of David
10. Ceva's Theorem
11. The Many Ways To Construct a Triangle
12. Concyclic Circumcenters: A Dynamic View
13. Concyclic Circumcenters: A Sequel