Department of Mathematics Education

J. Wilson, EMAT 6680

EMAT 6680 Assignment #9:
Pedal Triangle

Matthew Tanner

EMAT 6680

Write-up #9

July 23, 2004

## Pedal Circles– Pedal Point Collocated with the Circumcenter of a Triangle

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.

Figure 1

Recall that a unique circle may be inscribed on any distinct non-linear three points.  The circle inscribed on the three vertices of a triangle is called in Circumcircle.

Figure 2

The circle associated with any three non-linear points may be constructed by first finding the midpoints of two line segments joining any two pair of the three points.

Figure 3

Perpendicular lines intersecting the midpoints are guaranteed to intersect by the three points non-linearity.  The distance from the intersection (the center) to any of the three points is the diameter of the circle.

Figure 4

Observe that sides of a triangle are similarly bisected by perpendiculars that originate at the Circumcenter.

Figure 5

Situating the Pedal Point determining a Pedal Triangle at the Circumcenter has the immediate effect that the vertices of the Pedal Triangle bisect the sides of the principle triangle.

Figure 6

Because the vertices of the Pedal Triangle bisect the sides of the principle triangle, the Pedal Triangle is similar to the principle triangle with all sides being half the length.  The sides of the Pedal Triangle divide the principle triangle into four equivalent triangles.