By using Geometer's Sketchpad we can explore various relationships of Pedal Triangle. Given a point P and a triangle ABC, a Pedal Triangle is defined as the constructed triangle whose polygon vertices are the feet of the perpendiculars from P to the sides of triangle ABC. This point P is called the Pedal Point.
First we are given the Pedal Point P and a triangle ABC.
Next, extend the sides of triangle ABC and find the perpendicular lines from point P to these sides. Label the intersections with points R, S, and T.
Finally, connect these intersection points to form the Pedal Triangle RST.
Click here for a Geometer's Sketchpad script tool for the Pedal Triangle.
What would happen to the Pedal Triangle RST if point P were any point in the same plane as triangle ABC? Try and translate point P to observe the relationships between point P, the Pedal Triangle RST, and our given triangle ABC.
What will happen to the Pedal Triangle RST if the pedal point P is...
the centroid of triangle ABC? Watch the Pedal Triangle RST as point P moves to the centroid.
The Pedal Triangle now has all three vertices on the sides of our original triangle ABC. Regardless of the size or shape of triangle ABC, the centroid will always lie inside of triangle ABC (as long as triangle ABC is not degenerate!). Since this means point P will always lie inside triangle ABC as well, the vertices of triangle RST will remain along the sides of triangle ABC.
the incenter of triangle ABC? Watch the Pedal Triangle RST as point P moves to the incenter.
Triangle RST now has all three vertices on the sides of triangle ABC, similar to the centroid example above. Regardless of the size or shape of triangle ABC, the incenter will always lie inside of triangle ABC (as long as triangle ABC is not degenerate!). By experimenting with different sizes and shapes of triangle ABC, we can see that the vertices of triangle RST will remain on the sides.
the orthocenter of triangle ABC? Watch the Pedal Triangle RST as point P moves to the orthocenter.
Triangle RST now has all three vertices on the sides of triangle ABC, but do you notice something else? Each of the vertices also lie along the altitudes of triangle ABC. When triangle ABC is acute, these vertices R, S, and T lie on both the sides of triangle ABC, and on the altitudes. When triangle ABC is obtuse, its orthocenter lies outside the triangle. Consequently, the vertices of the Pedal Triangle will still lie on the altitudes, but they are no longer on the sides of triangle ABC. By experimenting with different sizes and shapes of triangle ABC, we can see that the vertices of triangle RST will remain on the altitudes of triangle ABC.
the circumcenter of triangle ABC? Watch the Pedal Triangle RST as point P moves to the circumcenter.
Similar to the orthocenter above, triangle RST has all three vertices on the sides of triangle ABC. This time, each of the vertices also lie along the perpendicular bisectors of triangle ABC. When triangle ABC is acute, these vertices R, S, and T lie on both the sides of triangle ABC, and on the perpendicular bisectors. When triangle ABC is obtuse, its circumcenter lies outside the triangle. Consequently, the vertices of the Pedal Triangle will still lie on the perpendicular bisectors, and they remain on the sides of triangle ABC. By experimenting with different sizes and shapes of triangle ABC, we can see that the vertices of triangle RST will remain on the perpendicular bisectors and the sides of triangle ABC.
on a side of triangle ABC? Watch the Pedal Triangle RST as point P moves to a side of triangle ABC.
When the Pedal Point P moves to one of the sides of triangle ABC, we can see P actually becomes on the of vertices of the Pedal Triangle. As shown by the image below, when point P is on the segment AC, point P and point T become the same point. The remaining vertices of the Pedal Triangle, S and R, will each lie on the other sides of triangle ABC when it is acute. If triangle ABC is obtuse, the remaining vertices of triangle RST can lie outside the triangle ABC. By moving point P to different points on each side of our triangle ABC, we can see that point P will be the same point as one of the vertices of triangle RST.
on a vertex of triangle ABC? Watch the Pedal Triangle RST as point P moves to a vertex of triangle ABC.
When the Pedal Point P moves to one vertex of triangle ABC, we can see all three vertices of triangle RST are collinear. This represents only one condition in which all three vertices of the Pedal Triangle RST are collinear. This is also called a degenerate triangle. This segment SR, shown below, is known as the Simson Line. By viewing the animation, we can see each time point P is moved to a vertex of triangle ABC, the vertices of the Pedal triangle RST will be collinear. By moving point P to different vertices of our triangle ABC, we can see that point P and all three vertices of triangle RST are collinear.
We can see in the example above that when the Pedal Point P is moved to a vertex of triangle ABC, the vertices RST of our Pedal triangle are collinear. In other words, our Pedal triangle is reduced down to a single line. This line is called the Simson Line. By experimenting with different positions of point P in Geometer's Sketchpad, we can see there are other ways to get the Simson Line.
In fact, there are several points around our triangle ABC where the Pedal triangle is reduced to this line. What relationships do these other points have to each other? What relationships do these points have to our original triangle ABC? Experiment with point P and try to figure it out. Move point P all around triangle ABC and note where P will have to be to get the Simson Line.
What do we already know about the Simson Line?
The Pedal triangle becomes the Simson Line when P is placed at each of the vertices of triangle ABC. The Pedal triangle becomes the Simson Line when P is placed around our triangle ABC in several places.
The Pedal triangle becomes the Simson Line when P is placed at each of the vertices of triangle ABC.
The Pedal triangle becomes the Simson Line when P is placed around our triangle ABC in several places.
What can we conclude?
So in order for the Pedal triangle to become Simson's line, P must hit each of the vertices of triangle ABC, and several points around our triangle ABC. Let's try experimenting with the circumcircle of triangle ABC. By definition, the circumcircle of triangle ABC passes through each vertex, and has the circumcenter as it's center. So, the circumcircle provides us with both, the vertices of triangle ABC and several points around it! Let's test our conjecture to see if it works for all points on the circumcircle. Constructing the circumcircle of triangle ABC, and move the Pedal point P around that circle. We can see that in fact, for all points on the circumcenter, our Pedal triangle will always become the Simson Line! Watch the Pedal triangle as point P moves around the circumcircle of triangle ABC.
So in order for the Pedal triangle to become Simson's line, P must hit each of the vertices of triangle ABC, and several points around our triangle ABC. Let's try experimenting with the circumcircle of triangle ABC. By definition, the circumcircle of triangle ABC passes through each vertex, and has the circumcenter as it's center.
So, the circumcircle provides us with both, the vertices of triangle ABC and several points around it! Let's test our conjecture to see if it works for all points on the circumcircle.
Constructing the circumcircle of triangle ABC, and move the Pedal point P around that circle. We can see that in fact, for all points on the circumcenter, our Pedal triangle will always become the Simson Line! Watch the Pedal triangle as point P moves around the circumcircle of triangle ABC.