The Simson Line
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Let ΔABC be any triangle, and let P be any point in the plane. Construct lines through P which are perpendicular to each of the sides of ΔABC. Let D, E, and F be the points of intersection of these lines with the lines to which they are perpendicular. We call ΔDEF the pedal triangle for pedal point P. Click here for a GSP file demonstrating the construction of a pedal triangle.
In this GSP file, move the pedal point P around until the pedal triangle is degenerate. That is, find a location of P such that the pedal "triangle" just looks like a line segment. Such a segment is called the Simson line. Once you have found such a location for P, move one of the blue circles (in the upper-left corner of your screen) to that location to mark it.
Repeat this process several times, until you feel comfortable making a conjecture about where P must lie in order for the pedal triangle to be degenerate. What property do the locations of your blue circles seem to share? Do they lie on a single line? On a parabola? What? Make a conjecture before clicking here to continue.
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