Problem: Investigate the pedal triangle for the different centers of the triangle.
Investigation/ Exploration:
I begin my exploration by discussing what a pedal triangle is. A pedal triangle is constructed by a triangle and an arbitrary point.
Then to continue the construction
of a pedal triangle one constructs perpendiculars from the arbitrary
point, called the pedal point to the three sides of the triangle.
Then by constructing the intersection
of the feet of the perpendiculars and the sides of my triangle
I obtain the vertices of my pedal triange. If I construct the
segments connecting the three intersections I obtain the pedal
triangle.
Notice that as I move the pedal
point inside the triangle around the triangle, and through the
sides of the triangle the pedal triangle changes shape becoming
at times degenerate.
Click HERE to see a gsp sketch where the pedal point moves randomly in the plane.
The pedal triangle of a point with respect to triangle ABC is triangle DEF whose vertices are the orthogonal projections of the point onto the sides of triangle DEF.
Now that I know what a pedal triangle is, what is the pedal triangle for the orthocenter of a triangle?
Notice that for an acute triangle where the orthocenter of the original triangle, triangle ABC is inside the triangle, the pedal triangle is also inside triangle ABC. Also notice that this triangle must be the orthic triangle. The orthic triangle is the triangle whose vertices are the endpoints of the altitudes of vertices of triangle ABC. These altitudes are concurrent at the orthocenter and necessairly go through the orthocenter. Therefore the feet of the perpendiculars of the orthocenter are the same as the endpoints of the altitudes of the vertices of triangle ABC. Therefore, the pedal triangle with respect to the orthocenter of a triangle is the orthic triangle. This occurs whether the orthocenter is inside the triangle or not.
Here is a gsp sketch of an obtuse triangle for which the orthocenter is located outside of triangle ABC.
What happens if the pedal point is the incenter of triangle ABC?
Here is a construction in gsp of the pedal triangle in which the pedal point is the incenter of triangle ABC.
Notice what happens if I construct the incircle of my triangle ABC.
Notice that when I construct the incircle of my triangle, the vertices of the pedal triangle of my incenter are the contact points of my incircle with the sides. This is called the intouch triangle. So the pedal triangle of the incenter of a triangle is the intouch triangle.
The incenter of a triangle is always located inside of the triangle.
What is the pedal triangle when my pedal point is the circumcenter of my triangle. The circumcenter is the intersection of the perpendicular bisectors of a triangle. notice that the feet of the perpendiculars through the circumcenter must be through the midpoints of the sides of the triangle.
Notice that the pedal triangle must be the medial triangle when the circumcenter is located inside the triangle.
What happens if the circumcircle is located outside of the circle instead of inside the circle.
The pedal triangle of a circumcenter when the circumcenter is outside of the triangle is still the medial triangle because the feet of the perpendiculars from the circumcenter to the sides of the triangle still go through the midpoints of the triangle. Notice that it still forms a triangle that is 1/4 the area of the large triangle. It also forms four congruent triangles inside of the large triangle.
What happens to the pedal triangle if the pedal point is the centroid of the triangle?
Notice that when I construct my pedal triangle with my pedal point being the centroid of the triangle I obtain a pedal triangle which looks like this:
Notice that if you have an obtuse triangle the pedal triangle formed by the centroid may be partly outside of the large triangle:
I was interseted in discovering what
happened when I constructed the centroid of the pedal triangle
and used that as the pedal point of another pedal triangle.
This is what my construction looks like:
Here is another picture of the pedal triangle with pedal point as the centroid of the second pedal triangle.
Theese are the discoveries I make about the pedal triangles for the four centers of a triangle.
Author & Contact:
Jennifer Shea