Pedal Triangles

By: Diana Brown

1a. Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

1b. Click here for a script tool for the general construction of a pedal triangle to triangle ABC where P is any point in the plane of ABC.

2. What if pedal point P is the centroid of triangle ABC?

The Pedal Triangle is inside triangle ABC  and the Pedal point is inside both triangles when P is the centroid.

3. What if . . . P is the Incenter . . . ?

The Pedal Triangle is another acute triangle that is inside triangle ABC when P is the Incenter.

4. What if . . . P is the Orthocenter . . . ? Even if outside ABC?

Inside ABC:

The Pedal Triangle stays in side triangle ABC and the pedal point is inside both triangles ABC and RST

Outside ABC:

The Pedal Triangle overlaps triangle ABC and the Pedal Point is outside both triangles ABC and RST.

5. What if . . . P is the Circumcenter . . . ? Even if outside ABC?

The Pedal Triangle is inside triangle ABC when the pedal point is inside triangle ABC.

The Pedal Triangle is inside triangle ABC even when the pedal point is outside triangle ABC.

6. What if P is on a side of the triangle?

If you drag the pedal point to one side of triangle ABC it becomes one of the vertices of triangle RST. The pedal triangle also always stays inside triangle ABC.

7. What if P is one of the vertices of triangle ABC?

If we take the pedal point to one of the vertices of triangle ABC the pedal triangle RST degenerates.  Thus the points of the pedal triangle are collinear.