Pedalling Triangles
By: Russell Lawless
Here are the steps in order to create a pedal triangle. Create a generic triangle ABC. Create a point P (this is your pedal point). Extend the each side of your triangle. Then create a perpendicular bisector for each of these extended sides that also goes through the point P. Create points at the intersection of the extended lines of the triangle and the perpendicular bisector. Next connect these three new points to create the pedal triangle.
Got all that? If not, here is a shortcut to a tool that is a construction of the pedal triangle: Pedal Triangle.
This is what it should look like:
While you are there explore what happens when the pedal point is on the side of a triangle ABC and on one of the vertices of the triangle ABC. When you attach the point to the segment, animate the point to see what happens.
Let's see what happens when the point is attached to a segment. What do you think?
at AB
at AC
at BC
We can say that when triangle ABC is acute, the vertices of the pedal triangle will lie on triangle ABC. The pedal point P will actually be one of those vertices. Here is a case for when triangle ABC is obtuse.
We can see that the pedal triangle lies outside triangle ABC when triangle ABC is obtuse. We also notice that not all of the vertices of the pedal triangle are on triangle ABC. So we can conclude that when a triangle is obtuse, the pedal triangle is not confined to the space inside of the original triangle and that at least one of the vertices will not be on any of the segments of the original triangle.
Now let's focus on when the pedal point P is attached to any of the vertices of triangle ABC. What do you think will happen?
at A
at C
at B
So we see that for the cases where the pedal point is attached to a vertex of triangle ABC, it creates a line that goes along the altitude of the triangle. Other names for this specific line are the degenerate triangle or the Simson Line. What about obtuse triangles?
We see that the middle of the three lines (line HI) creates a line that goes along the altitude of triangle ABC. So we can conclude that this is also true for obtuse triangles. Therefore, when the pedal point P is on the vertex of triangle ABC, it shows the altitude of the triangle.
So depending on the location of P, we are able to get multiple forms of the pedal triangle.