**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 P.Pedal Point
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. |