Pedal Points and Triangles
by Morgan Guest
Given a triangle ABC and Pedal Point P anywhere on the plane, a Pedal Triangle can be constructed. Extend lines AB, BC, and AC and construct perpendicular lines from P to AB, BC, and AC. The vertices of the Pedal Triangle are the intersection points of the perpendicular lines with the (sometimes extended) lines AB, BC, and AC. An example of a Pedal Triangle can be seen below. The left image shows a Pedal Triangle where the Pedal Point is outside the triangle. The right image shows a Pedal Triangle where the Pedal Point is inside the triangle. For more exploration with the gsp file, click here.
The Pedal Point P can be anywhere on the plane, so lets explore what happens if we place the Pedal Point at the same place as.....
the Circumcenter -- As pictured below, the if the Pedal Point is the circumcenter of the original triangle, the Pedal Triangle is the same as the medial triangle. The vertices of the medial triangle are the three midpoints of the sides of the original triangle. The circumcenter is the concurrent point of the perpendicular bisectors of the sides of the original triangle. The perpendicular bisectors intersect the sides of the original triangle at the midpoint of the sides. Therefore, the Medial Triangle is the same as the Pedal Triangle if the Pedal Point is the circumcenter. For further exploration, click here.
the Orthocenter -- As pictured below, if the Pedal Point is the same as the orthocenter of a triangle ABC, the Pedal Triangle is the same as the orthic triangle. The vertices of the orthic triangle are the intersection points of the altitudes with the sides of the original triangle. These are the same vertices of the Pedal Triangle when the Pedal Point is the orthocenter. For further exploration, click here.
the Incenter -- This is one of the more interesting findings. If the Pedal Point is the same as the incenter of the original triangle ABC, then P is the circumcenter of the Pedal Triangle. The incenter is equidistant from the sides of the original triangle. Distance is measured with perpendicular lines, so the incircle is tangent to the original triangle at the intersections of the perpendiculars from the incenter to the sides of the original triangle. Therefore, the Pedal Point is equadistant from the vertices of the Pedal Triangle, which is the definition of the circumcenter. For further exploration, click here.
the center of the 9 point circle -- As you can see below, there is nothing unique about the Pedal Point being the center of the 9 Point Circle except for when the 9 Point Circle is constructed from an equilateral triangle. The orthocenter and circumcenter are the same point for an equilateral triangle, so they are also the center of the 9 point circle. The Pedal Triangle of an equilateral triangle will lie on the 9 Point Circle, otherwise, the Pedal Triangle does not lie on the 9 Point Circle.
The picture above shows that when the Pedal Point is the center of the 9 point circle of an equilateral triangle, the Pedal Triangle lies on the 9 Point Circle. For further exploration with the Pedal Triangle of an equilateral triangle click here.
The picture above shows that when the Pedal Point is the center of a 9 point circle, the Pedal Triangle does not have any points on the 9 Point Circle. For further exploration with the Pedal Point the same as the center of any 9 Point Circle click here.
