Neuberg's-Theorem: The Third Pedal Triangle of a Triangle Using Sketchpad, construct ΔABC. Choose any point D, and construct the first pedal triangle ΔEFG with respect to D. (Use the procedure of Steps 1 and 2 from the Discovery Unit above-Pedal Triangles and Simpson's Line.) From D, construct the pedal triangle ΔHIJ of ΔEFG, and finally, from D, construct the pedal triangle ΔKLM of ΔHIJ (the second and third pedal triangles of ΔABC with respect to D). If you know how, create a Script for the process of constructing the first pedal triangle, selecting A, B, C, and D, in order, and use this for the second and third pedal triangles.
(a) Drag the vertices A, B, and C in such a way that you make ΔABC equilateral (or as close as you can make it). Did you observe anything peculiar happening to ΔKLM?
(b) Make ΔABC a distinctive right triangle, say with one rather long leg. What happened this time? In both cases, drag D and observe the effect.
NOTE: The theorem being demonstrated here is due to J. Neuberg. Its proof may be found in Coxeter and Greitzer, Geometty Revisited, p. 24.
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