Pedal Triangles

Let's construct a pedal triangle. First we need a triangle; we'll construct it by using lines rather than just segments. Label the vertices A,B,and C. Now pick a point, any point in the plane, and label it P.

Now we construct perpendiculars from P to each side of the triangle. Label the instersections of the perpendiculars and the sides of the triangle, R,S, and T.

 

Now connect those intersection points, that is, R,S, and T. This forms the Pedal Triangle for Pedal Point P.

Click here to find a GSP file. Drag Point P around to different places in the plane to see what happens to the pedal triangle.


Let's see what happens when pedal point P is the centroid G of triangle ABC. Recall that the centroid of a triangle is the intersection of the three medians, segments from the vertices to the midpoint of the opposite sides.

 

Click here to find a GSP file. Drag any vertex to see what happens to the pedal triangle RST. Notice that the pedal triangle is always inside the original triangle.


Now let's look at the pedal triangle if pedal point P is the incenter I of triangle ABC. Remember that the incenter is the point in a triangle that is equidistant from the three sides. It lies on the angle bisector of each angle of the triangle. Construct perpendicular lines from I to each side of triangle ABC and connect the intersection points to form triangle RST.

 

Click here to see a GSP sketch of the pedal point as the incenter of triangle ABC. Again the pedal triangle is inside the original triangle.


What if the pedal point is the oorothcenter of triangle ABC? Recall that the orthocenter, H, of a triangle is the intersection of the three lines containing the altitudes. Those are segments drawn from each vertex perpendicular to the opposite side, which may have to be extended.

 

 

 

 

Click here to manipulate a GSP file of this sketch. Drag the sketch by A,B, or C. What happens to the pedal triangle here? What if the orthocenter, H, is outside triangle ABC? Did you find that the pedal triangle disappears when H is outside ABC? Were there any places where the pedal triangle becomes a Simson line? Did you check the points where H is on a vertex?


Finally, let's look at the situation when the pedal point is the circumcenter of triangle ABC. The circumcenter C is the point equidistant from the vertices of a triangle. It is the intersection of the perpendicular bisectors of the three sides of a triangle.

 

Click here for a GSP sketch which can be manipulated by A,B, or C(the vertex). Notice that when C (the circumcenter)is inside triangle ABC the pedal triangle is always visible. And even when the circumcenter is outside, the pedal triangle is evident except at the place where both triangle collapse on top of each other into a Simson line.


Further investigations can be done with the pedal point. Try P as the center of the nine-point circle for triangle ABC. What if P is on a side of the triangle? A vertex? Try tracing lines constructed along the sides of the triangles. See Assignment 9 under EMT 668 on Jim Wilson's Home Page for further suggestions.


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