The Pedal Triangle (continued)
By
Sharon K. OÕKelley
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;
thereforeÉ
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; thereforeÉ
It can be concluded then
through substitution thatÉ
Using the angle addition
postulate, the following relationships can be establishedÉ
Since these two angles
share common angle DPC, angle DPC can be subtracted from both thus yieldingÉ
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 establishedÉ
Using substitution
establishes thatÉ
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 PtolemyÕs Theorem? Go here for the answer.