In this exploration we will look at how a pedal triangle is constructed and the relationship between the pedal triangle and some of the different centers of a triangle.
A pedal triangle is the triangle that is formed when a the perpendiculars of the legs of the given triangle pass intersect through a given point P. This given point P is called the pedal point. The picture shows an example where the pedal point is outside of the triangle and the legs of the triangle were extended. The perpendiculars are colored red to better understand the construction.
We will now look at the pedal triangle with regard to the orthocenter and circumcenter. These two centers are located within a triangle if the triangle is acute, on the triangle if it is a right triangle, and outside of the triangle if it is an obtuse triangle.
Let us first examine the orthocenter, which the most interesting case occurs when the given triangle is a right triangle. The orthocenter is located at the vertex of the triangle where the right angle is located. If we let the pedal point, P, be the same point as the orthocenter the pedal triangle becomes a line and divides the right triangle in half.
Next we will look at the orthocenter, and what occurs when the pedal point is located at the orthocenter of the given triangle. First case will be an acute triangle.
Looking at the ratios of the areas of the original triangle and the pedal triangle, it is easy to see that the pedal triangle is one fourth of the given triangle and it actually divides the given triangle into four congruent triangles.
Now let us examine a right triangle and what happens when the pedal point, P, is located at the circumcenter.
Again the ratio is four and the pedal triangle is one fourth the given right triangle.
Now let us examine an obtuse triangle and what happens when the pedal point, P, is located at the circumcenter.
Again the ratio of the areas is four. As you can see in all cases when the pedal point is located at the circumcenter the ratio remains at the constant four and the pedal triangle divides the given triangle, ABC, into four congruent triangles.
Note: This is NOT a proof of the above conjectures. These are just observations.
Click here to download a GSP sketch that has a constructed pedal triangle and four centers (centroid, orthocenter, incenter, and circumcenter).