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
the three points D,E, and F that are the intersections. Triangle
DEF is the Pedal Triangle for Pedal Point G.
How do you construct a pedal triangle?
Construct a triangle and label the vertices A, B, and C. Pick a random point, G, outside to the triangle and extend segments AB BC, and AC. Construct perpendicular lines that passes through point G and intersects each extended segment. The points of intersection will form a triangle-the pedal triangle (DEF) of pedal point G.
Click here to see pedal triangle
What if the pedal point, P or G, is the orthocenter? What if the pedal point is outside of ABC?
When triangle ABC is acute, the pedal point, G, lies inside the triangle ABC. Notice that the vertices of triangle DEF lie on the altitudes and the sides of ABC.
When triangle ABC is obtuse, the pedal point, G, lies outside of the triangle ABC. Notice that the vertices of triangle DEF still line on the altitudes of the triangle.
Thus given any triangle, acute or obtuse, the vertices of the pedal triangle will lie on the altitudes of the original triangle.
What if the pedal point, G, is the centroid of triangle ABC?
When point G is the pedal point and the centroid of any given triangle, the vertices of the pedal triangle, DEF, lie on the sides of triangle ABC. Also, the pedal triangle will always be inside of triangle ABC. Thus triangle ABC can take any shape or form and this will remain true.
What if the pedal point G is on a side of the triangle of ABC?
When G, the pedal point, is on a side of triangle ABC, it is one of the vertices of the pedal triangle. Above G = D, the pedal points lies on segment AC, and the other vertices of the pedal triangle lie on the other sides of triangle ABC. If G, the pedal triangle lies on a side of triangle ABC and if triangle ABC is acute, then the pedal triangle is inside of triangle ABC. What happens if triangle ABC is obtuse and the pedal point, G, lies on a side? Explore