Sinje J. Butler

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**Pedal Triangles**

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The following is triangle **ABC**. Point **P **is the pedal
point (which could be any point in the plane). From point **P,
**lines have been constructed that are
perpendicular to each side of the triangle. These intersections are labeled **R, S **and **T**. **RST** is the pedal triangle**.**

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Please click here for a **GSP Demonstration**
that shows the pedal triangle for **P** at different places
throughout the plane.

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The following will explore some
correlations that can be made between the pedal triangle and others triangles
dependent upon the location of **P.**

If pedal point **P **is
the orthocenter of triangle **ABC** the pedal triangle is the orthic triangle. This is true for the case when the
orthocenter is inside triangle **ABC** and this is the case when the orthocenter is outside the
triangle. Please click here for a **GSP Demonstration**.

This occurs because the
orthocenter of **ABC **lies on each of the three altitudes of this triangle. When **P** is the
orthocenter it also lies on each of the three atlitudes. Therefore, **P**
is perpindicular to each side at the vertices of the orthic triangle and thus
the orthic triangle is the pedal triangle.

If the pedal point **P **becomes
the circumcenter, the pedal
triangle is the medial triangle. This occurs because when **P**
is placed on the circumcenter it lies on the perpendicular bisectors of the
sides of triangle **ABC****. **So
**R, S **and **T**** **are
perpendicular to the original triangle and they lie on the three midpoints of
the three sides, thus when these segments are connected they are the medial
triangle. Please click here for a **GSP Demonstration** of **P** as the circumcenter.
Notice, that if the circumcenter is outside of the triangle **ABC**, when
**P**** **lies on top of the circumcenter, it is the medial triangle.

__Other Observations__

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Please click here for **GSP
Demonstration **to move **P** throughout the plane.(PedCircmcir1) Notice that as **P** is moved around it appears there are positions of **P** for which the vertices of the pedal triangle are
collinear. Where does this occur
and why?

The next **GSP Demonstration ** appears to show that if the circumcirle
of triangle **ABC** is
constructed and point P** **is merged onto the circumcircle, the vertices of the
pedal triangle are collinear for any point on the circumcircle.(PedCircmcir2)