The Pedal Triangle (continued)

By  Sharon K. O’Kelley

 

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

 

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.

 

Figure 5

 

 

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; therefore…

 

 

 

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.

 

 

Figure 6

 

 

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

 

 

It can be concluded then through substitution that…

 

 

Using the angle addition postulate, the following relationships can be established…

 

 

Since these two angles share common angle DPC, angle DPC can be subtracted from both thus yielding…

 

 

 

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

 

Figure 7

 

 

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

 

 

 

Using substitution establishes that…

 

 

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.

 

What connection does the Simson Line have to Ptolemy’s Theorem? Go here for the answer.

 

 

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