LetŐs begin by constructing triangle ABC and its pedal triangle for an arbitrary point P, in the plane. The pedal triangle is formed by connecting the intersections of the perpendicular lines from P to each of the sides of the triangle.
Notice that P is inside the triangle, letŐs see what happens if P is outside the triangle.
It looks as if when P is inside the triangle, then the pedal triangle is inside also. Similarly, if P is outside the triangle then the pedal triangle will also be outside. LetŐs continue to see if my conjecture is correct.
Now, letŐs see what happens if P is a vertex of the triangle.
LetŐs try the other vertices before I make a conjecture.
When P is a vertex, the pedal triangle collapse. I wonder why?
What about when P is on one of the sides of the triangle.
P then becomes one of the vertices of the pedal triangle. This is true for all cases.
Now, letŐs try some other locates for P.
What about when P is the centroid?
The pedal triangle appears to be equilateral. I checked to see if the vertices of the pedal triangle intersect the midpoints of the triangle, it does not. But it was pretty close.
What about when P is the incenter?
I donŐt see any special relationships with this. The pedal triangle is no longer equaliteral and its vertices are not in any significant location. LetŐs try the orthocenter.
The pedal triangle becomes the orthic triangle. This is true because the orthocenter is the intersection of the altitudes of triangle ABC. The three points used to form the orthic triangle are feet of the altitudes. So the vertices of the orthic triangle are formed from the perpendicular from the orthocenter to each side. Since P is located at the orthocenter, the vertices of the pedal triangles are the same as the vertices of the orthic triangle.
Lastly, letŐs see what will happen if P lies on the circumcenter of the triangle.
The pedal triangle becomes the medial triangle in this case. This is true because the vertices of the medial triangle are the midpoints of the sides of the triangle. The circumcenter is perpendicular to each side through its midpoint. Therefore, when P is the circumcenter, the vertices of the pedal triangle would be the midpoints of the sides of the triangle.