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*A Special Case of the Pedal Triangle*

*By Sharon K. OŐKelley*

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*Introduction*

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**This presentation provides
an overview of the degenerate case of the Pedal Triangle, explores the resulting
Simson Line, and provides a connection between the Simson Line and PtolemyŐs
Theorem. The goal here is to demonstrate for an advanced high school math
student or teacher how various mathematics topics can be connected together
– i.e., to show the relationships that can be established through
exploration and explanation. Some of the topics addressed here are altitudes of
triangles, circumcircles, cyclic quadrilaterals, and trigonometry. This
presentation is centered around constructions and observations made from the
use of GeometerŐs Sketchpad.**

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*The Pedal Triangle*

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**Given a point P either
inside or outside a triangle, a Pedal**

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****Figure
1
Figure
2**

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*The Degenerate Case of the Pedal Triangle*

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**Consider figure 3 in which
circumcircle O has been constructed for the given triangle. Note that point P
is off the circumcircle and the vertices of triangle ABC are noncollinear.**

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**Figure 3**

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**Using GeometerŐs
Sketchpad, point P can be merged with the circle. When this occurs, the Pedal
Triangle collapses or becomes degenerate which means the vertices of A, B, and
C become collinear thus creating the Simson Line. In figure 4, line AC
containing point B is the Simson Line. **

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**Figure 4**

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**No matter where P falls on the circumcircle, the
Pedal Triangle remains collapsed as the Simson Line.**

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**To download an animation of the Simson Line using GeometerŐs
Sketchpad, click here.**

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**For an explanation behind
the creation of the Simson Line, go here.**

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