Assignment 9: Pedal Triangles
Jason P. Pickhardt
The following questions were explored about pedal triangles:
1. What if pedal point P is the centroid of triangle ABC?
2. What if . . . P is the incenter . . . ?
3. What if . . . P is the Orthocenter . . . ? Even if outside ABC?
4. What if . . . P is the Circumcenter . . . ? Even if outside ABC?
5. What if P is on a side of the triangle?
6. What if P is one of the vertices of triangle ABC?
To answer the first four questions, it is best to use Geometerís Sketch Pad. The associated constructions and discussions can be found here.
When exploring question 5, we find that there are a multitude of pedal point possibilities along the sides of a triangle. Consider triangle ABC shown below in dark blue. We can construct pedal point P on segment AB. When we construct lines perpendicular to each side and point P (by definition of a pedal triangle) we realize that point P is also a vertex R or pedal triangle RST shown below.
In GSP it is possible to vary point P,R along segment AB to construct many pedal triangles. These triangles are not of immediate interest to us, however when Point P,R varies to either vertex A or vertex B we do† have something of interest.
The lines in red emanating from vertex A and B to the side of the triangle opposite them respectively are known as Simson Lines. A Simson Line is one where the vertices of a triangle are collinear and thus creating what is known as a degenerate triangle. We can see that this is true whenever we have a pedal point on a side of a triangle that varies to the any vertex of that triangle. Plotting pedal points Pí and Píí we create pedal triangles PíRíTí and PííRííTíí shown below. Dragging these points to the vertices of their respective segments gives us the Simson Lines associated with the pedal points shown below.
Notice that we some similarities between the triangles having Simson Lines. The Simson Lines formed by pedal triangle RST and RííSííTíí both at vertex A form the same Simson Line. This is that same for RST and RíSíTí at vertex B and RíSíTí and RííSííTíí at vertex C. To see the pedal points vary in GSP click here and use the animation buttons to see the pedal triangles become Simson Lines.
Through the pedal triangle constructions attached and those previously discussed we can see how triangles can be related. In the discussions on the GSP file we brought up such triangles as the medial triangle and the orthic triangle. These constructions are related to the centers of triangles and this was seen through our exploration.