__Assignment
#9
__

__Pedal
Triangles__

A pedal
triangle is created from any triangle ABC and a point P in the same plane.

To construct
the pedal triangle construct perpendicular lines to the sides of the triangle
ABC (the sides may have to be extended) through point P.

The
resulting intersections with the perpendicular lines and the triangle sides,
points R, S, and T, are the vertices of the new pedal triangle RST for point P.

What happens
when point P is the Orthocenter?

Conjecture: When point P is the orthocenter, the pedal
triangle is also the orthic triangle.

Proof: The pedal triangle is created by constructing
a perpendicular from point P to the sides of the triangle. The orthocenter is the common intersection of
the altitudes of a triangle.

The orthic triangle is the triangle formed by the intersection
of the altitude and the triangle sides.

When P is
the orthocenter, the perpendicular lines projected from P lie
o the altitude and, therefore, share the same intersection with the sides of
the triangle. Since these common
intersections of altitude and the triangle sides also form the orthic triangle, the conclusion can be made that when P is
the orthocenter, the pedal triangle created is also the orthic
triangle of triangle ABC.