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1a. Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

Example, when Pedal Point is outside the triangle and when it is inside the triangle

Notice that when the pedal point P is outside of the triangle then the pedal triangle is also outside of the triangle and when P is inside of the triangle so is the pedal triangle.

1b. Use GSP to create a script tool for the general construction of a pedal triangle to triangle ABC where P is any point in the plane of ABC. Click here for script tool and to move the pedal point P.

2. What if pedal point P is the centroid of triangle ABC?

If P is the centroid, then the pedal triangle is inside the triangle. Since the centroid is always inside the triangle, it makes sense that the pedal triangle will also stay inside the triangle is this case.

3. What if . . . P is the incenter . . . ?

If P is the incenter then the pedal triangle is inside the triangle...and P becomes the circumcenter of the pedal triangle!!

4. What if . . . P is the Orthocenter . . . ? Even if outside ABC?

If P is the orthocenter then the pedal triangle becomes the Orthic Triangle!!

5. What if . . . P is the Circumcenter . . . ? Even if outside ABC?

If P is the circumcenter, then the pedal triangle becomes the Medial Triangle. Each side of the pedal triangle represents the midsegments of the original triangle despite when P is inside or outside the triangle.

6. What if . . . P is the Center of the nine point circle for triangle ABC?

If P is the center of the nine point circle then the pedal triangle is inside of the triangle and P is the circumcenter of the pedal triangle!!

7. What if P is on a side of the triangle?

If P is on the side of the triangle, then P becomes a vertice of the pedal triangle. Despite the type of triangle, two vertices of the pedal triangle will always lie on the original triangle!!

8. What if P is one of the vertices of triangle ABC?

If P is one of the vertices, then the pedal triangle becomes the altitude of the triangle. This makes sense because the pedal triangle is constructed by perpendiculars of the sides of the original triangle and the altitude is the perpendicular to a side from a vertice. As you can see in an obtuse triangle the pedal triangle/altitude is outside of the triangle. In the acute triangle it is inside the triangle. What do you think will happen in a right triangle.....