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*The Pedal Triangle (continued)*

*By
Sharon K. OÕKelley*

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*Why is the Simson Line Created when P is on the
Circumcircle of the Original Triangle?*

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**Consider figure 5. Recall
that points A, B, and C are collinear and are also the vertices of the degenerate
pedal triangle with P as its pedal point. Here, we will explain why A, B, and C
are collinear and thus create the Simson Line.**

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**Figure 5**

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**In figure 5, the original
triangle has been labeled as DFE. Notice that points D, P, F, and E can be joined
to create a cyclic quadrilateral meaning that the quadrilateral is inscribed on
the circle. In cyclic quadrilaterals, opposing angles are supplementary;
thereforeÉ**

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**Next, consider triangle
AEC which has its own circumcircle S as shown in purple in figure 6. Note that
P lies on circle S as well as circle O.**

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**Figure 6**

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**Circle S contains the
cyclic quadrilateral APCE. Since angle E and angle APC are opposing angles of
the cyclic quadrilateral, they are also supplementary; thereforeÉ**

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**It can be concluded then
through substitution thatÉ**

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**Using the angle addition
postulate, the following relationships can be establishedÉ**

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**Since these two angles
share common angle DPC, angle DPC can be subtracted from both thus yieldingÉ**

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**Next, consider that there
are two other circumcircles involving point P as shown in figure 7.**

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**Figure 7**

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**Because quadrilaterals
PFCB and PADB are cyclic quadrilaterals on their respective circles, the
following relationships can be establishedÉ**

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**Using substitution
establishes thatÉ**

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**Because these two angles
are equal, they can be verified as vertical angles; therefore, points A, B, and
C are collinear making the pedal triangle degenerate and thus establishing the
Simson Line.**

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**What connection does the
Simson Line have to PtolemyÕs Theorem? Go here for the answer.**

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