A ** pedal triangle** is constructed
as follows:

Let triangle ABC be any triangle. Let P be any point in the plane. Construct perpendiculars to the sides of ABC (extended if necessary). Locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

Above, **D RST** is the pedal triangle of **D ABC **when
**P** is the pedal point. The dashed circle is the circumcircle
of **D ABC
**with its center at **C** (NOT the
vertex C). Click on the picture to manipulate the location of
the pedal point **P**. Make the following observations:

1.If the pedal point is outsideD ABCand in the interior of an angle (if you imagine that the sides which form the angle at a vertex are rays extending away from the vertex) ofD ABC,then at least one vertex of the pedal triangleD RSTlies on a side ofD ABC.Specifically, the side is the third side (not used to form the angle).

2.If the pedal point is inside theD ABC, then the vertices of the pedal triangle lie on the sides ofD ABC. That is, the pedal triangle is insideD ABCwith its vertices on the sides ofD ABC.

3.If the pedal point is the circumcenter ofD ABC, then the vertices of the pedal triangle are the midpoints of the sides ofD ABC.

4.If the pedal point is on a side ofD ABC, then the pedal point is in fact one of the the vertices of the pedal triangle.

5.If the pedal point is one of the vertices ofD ABC, then the pedal triangle has no area. That is, the verticesR, S, Tof of the pedal triangle are collinear. In other words, the pedal triangle is a segment.

6.If the pedal point lies on the circumcircle ofD ABC, then the pedal triangle is again a segment since its vertices are collinear. Push theANIMATEbutton to observe the pedal triangles whose pedal points lie on the circumcircle.

In **5 **and **6**, the pedal triangle
is a segment. This special segment is called the ** Simson
Line**.

If the **D
ABC** is *acute*
or* right*, then notice that there are times when the Simson
Line is actually one of the sides of **D ABC**.
This occurs when the pedal point is the intersection of the circumcircle
and the ray which begins at a vertex and passes through the circumcenter.

That is, let **ray AC **intersect the circumcircle
at **X, ray BC **intersect the circumcircle at** Y**, and**
ray CC **intersect the circumcircle at **Z. **Then if **X,
Y, **and** Z **are pedal points, each of their respective
Simson lines (degenerate pedal triangles) would coincide with
the side opposite the vertex of that point's respective ray.

However, if the triangle is *obtuse* with
the above conditions, then the 3 Simson lines still coincide with
the sides of **D ABC**. However, only one of
the Simson lines is the same length as its respective side. Click
on the picture and make the triangle obtuse.

For which pedal point (

X, Y,orZ) is the Simson line a side?

That is:The point which lies in the interior of the obtuse angle.if

< Ais obtuse, then the Simson triangle for pedal pointXwill be sideBC.if

< Bis obtuse, then the Simson triangle for pedal pointXwill be sideAC.if

< Cis obtuse, then the Simson triangle for pedal pointXwill be sideAB.