Let P be any point inside triangle ABC as shown. Draw the lines PA, PB, and PC. These lines meet the circle inscribed in the triangle at points X, Y, and Z, respectively. The inscribed circle touches the sides of the triangle at points D, E, and F, as shown. Then, no matter where point P is chosen, the lines DX, EY, and FZ will always meet at a point (labelled Q in the figure).
If you have a QuickTime Movie Viewer, you can see an animated version of our logo (354K). This animation shows point P moving around the triangle and you can see that lines DX, EY, and FZ remain concurrent at all times. The animation was produced using Geometer's Sketchpad (version 2.1) and flattened with FastPlayer (version 1.1).
There are many generalizations of this pretty result. The result was discovered by Stanley Rabinowitz in 1990. The details surrounding its discovery is a long story.
Stanley Rabinowitz, "Problem 1364", Mathematics Magazine. 64(1991)60.
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