Isogonal image of the circumcircle. What is it?

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Isogonal image of the circumcircle: What is it?
A Mathematical Droodle

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Explanation

Let P be a point in the plane of ΔABC. Reflect the lines PA, PB and PC in the bisectors of angles A, B, and C, respectively. In most cases, the lines thus obtained are concurrent. The point Q of concurrency is known as the isogonal conjugate of P.

In fact, the three lines always intersect, not just in most cases, although some cases do appear exceptional. When P happens to lie on the circumcircle of ΔABC, its isogonal conjugate lies on the line at infinity. When P lies on the circumcircle, the reflections of PA, PB and PC become parallel, i. e., intersect at infinity.

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.

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What if applet does not run?

40601115

Isogonal image of the circumcircle: What is it? 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.

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.

Isogonal image of the circumcircle: What is it?
A Mathematical Droodle