LetDABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing the perpendiculars to the sides of ABC (extended if necessary) locate three points R, S and T that are the intersections. DRST is the Pedal Triangle for Pedal Point P. Here’s a picture:
In this exploration, I used several tools from my GSP 4.0 library to explore some special cases of pedal point locations and the resulting pedal triangles. Those tools are replicated in the sketches referenced below if you’d like to validate or explore these conjectures further. To go directly to a case you’re interested in, click one of the links below:
This sketch shows separate orthic and pedal triangles when the pedal point is not concurrent with the orthocenter. When the pedal point is moved over the orthocenter, the two triangles coincide:
This finding is not very surprising, given the definitions of the orthocenter, orthic triangle and pedal triangle. The pedal triangle is formed by connecting the feet of the altitudes from an arbitrary pedal point P to the (extended) sides of a triangle (ABC). The orthocenter is the intersection of the altitudes from any vertex to the opposite side of the triangle. The orthic triangle is formed by connecting the “feet” of those altitudes. By moving P to the orthocenter’s “spot”, we’ve merely moved the “arbitrary” pedal point to a location where the perpendiculars have meaning ? as the altitudes of DABC.
This finding is a little more surprising, but looking at the definitions of incenter, incircle and pedal triangle makes it obvious. The incenter is the intersection of the angle bisectors of DABC and is the center of the incircle, the inscribed circle. As such, the incenter is equidistant from each of the three sides of DABC. The perpendiculars from the incenter to the sides of DABC form intersections with the sides of DABC that are the points of tangency. By moving the pedal point P to the incenter’s location, we have once again, removed the “arbitrariness” of P and placed it in a location where the altitudes have meaning as the radii of the incircle.
You can see separate pedal and medial triangles. When P is moved to the circumcenter, the triangles overlay one another as seen here:
Here is a picture when the circumcenter is outside the triangle:
Does this finding fall out as easily from the definitions as the others did? The circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle. It’s also the center of the circumcircle. The medial triangle is the triangle formed by connecting the midpoints of the sides of the triangle. When the pedal point is moved to the circumcenter’s location, the perpendiculars from P to the sides of the triangles are the perpendicular bisectors of the sides, which by definition pass through the midpoints of the sides. It follows that when P is the circumcenter, the pedal triangle is the medial triangle.