The Simson Line
by Emily Kennedy

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.

Next (The Simson Line) > >

Return to my page
Return to EMAT 6680 page