Pedal triangle

Encyclopedia : P : PE : PED : Pedal triangle

A triangle in red, the perpendiculars from a point P in green, and the obtained pedal triangle in blue.

In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.

More specifically, consider a triangle ABC, and a point P. Drop perpendiculars from P to the three sides of the triangle (these may need to be produced, i.e., extended). Label L, M, N the intersections of the lines from P with the sides BC, AC, AB. The pedal triangle is then LMN.

The location of the chosen point P relative to the chosen triangle ABC gives rise to some special cases:

The case when P is on the circumcircle, and the pedal triangle degenerates into a line (blue).

If P is on the circumcircle of the triangle, LMN collapses to a line. This is then called the pedal line, or sometimes the Simson line after Robert Simson.

See also

• Simson line

External links

• [Simson's line] on PlanetMath
• [Mathworld: Pedal Triangle]
• [Java Applet of the Perpendiculars]
• [Simson Line]
• [Pedal Triangle and Isogonal Conjugacy]

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