Symmedian in a Right Triangle

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

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Buy this applet
What if applet does not run?

The applet suggests a simple fact that, in a right triangle, the symmedian to the hypotenuse coincides with the altitude from the right angle.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Buy this applet
What if applet does not run?

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.

Symmedian

49552041