by

Rich Moushegian

Write-Up #9: Investigation of Pedal Triangles

Definition of a Petal Triangle: Let triangle ABC be any triangle with point P in the plane. Then the triangle formed by constructing perpendiculars to the sides of ABC (extended lines if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle as shown below.

For a Pedal Triangle script, select three points to define a triangle and an outside point.

If pedal point P is collocated with the centroid K of the given triangle (ABC), then the pedal triangle is interior to the given triangle. Note that the upper side of the pedal triangle (red) is not parallel to the base of the given triangle (black).

If pedal point P is collocated with the incenter, I of a triangle, then the pedal triangle is also interior to the given triangle.

If pedal point P is collocated with the orthocenter, H of a triangle, then the pedal triangle is interior to the given triangle. Note that the orthocenter is constructed the same way (note the construction lines) as the pedal triangle when the pedal triangle is inside the given triangle.

When, due to construction, the orthocenter is located outside the triangle, and the pedal point is collocated with the othocenter, then the pedal triangle exceeds the bounds of the original triangle, but the triangle is consistent with the othocenter, as expected.

If pedal point P is collocated with the incenter C, then the pedal triangle will locate inside the triangle with sides parallel to the original triangle because the vertices coincide with the midpoints of the original triangle.

However, if the pedal point P traces the circumcircle of the original triangle, then the pedal triangle is reduced to a line called the Simson Line. (Click here for animation of the trace about the circumcircle.) Of course if the point P is located at a vertex of the original triangle, the pedal triangle will be a Simson Line because the vertices are on the circumcircle.

Finally if the pedal point P moves along the circumcircle and we trace the Simson Line, we will get the following deltoid figure. (Click here for an animation of the following figure.)