Circles and Reflections in a Triangle

Circles and Reflections in a Triangle: What is this about?
A Mathematical Droodle

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Explanation

I found the following fact in D. Wells' Curious and Interesting Geometry (Penguin Books, 1991, p 258.)

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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Let P be a point in the plane of ΔABC. Reflect P in the side lines of the triangle and let K_a, K_b, and K_c denote the reflections in sides BC, AC, and AB, respectively. Then the circumcircles of four triangles ABC, AK_cK_b, BK_aK_c and CK_bK_a are concurrent.

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Circles and Reflections in a 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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Circles and Reflections in a Triangle: What is this about?
A Mathematical Droodle