Assignment 9

Pedal Triangles

by Jeff Hall

Problem 1a

Draw any Triangle ABC. Point P is any point in the plane. Construct the perpindiculars to the sides

of ABC (extended if necessary) and label the three intersections R, S, and T.

This Triangle RST is called the Pedal Triangle for Pedal Point P.

Problem 1b

A pedal triangle can be formed from any triangle where P is any point in the plane of ABC.

A GSP tool, titled PedalTriangle, has been created to assist in the creation of these triangles.

As in Assignment 5, click on the link above and choose the Pedal Triangle tool to use it.
Problem 2

What happens when P is on the centroid of the triangle?

You get this:

Notice that the pedal triangle is similar to a medial triangle,

except that the vertices of are not on the midpoints of Triangle ABC.

If Triangle ABC is an equilateral triangle, then Triangle RST becomes

a medial triangle.

Problem 3

Now what happens when the Point P is on the incenter of Triangle ABC?

Again, it created a triangle similar to the medial triangle of Triangle ABC.

Since the Incenter and Centroid are the same point in an equilateral triangle,

the Pedal Triangle is a medial triangle when P is on I.
Problem 4

What if P is on the Orthocenter?

As you can see from the rays, the pedal triangle is the orthic triangle when P is on H.

But what happens when H is outside the lines of Triangle ABC?

The pedal triangle becomes the orthic triangle of Triangle HBC.
Problem 5

What if P is on the Circumcenter?

The pedal triangle is now the medial triangle for Triangles ABC...and it's not equilateral!

But what happens when the Circumcenter lies outside of Triangle ABC?

The pedal triangle remains the medial triangle despite the circumcenter lying outside.

Problem 7

What if P is on a side of the triangle?

Is Triangle RST always contained in Triangle ABC when P is on a side?

As you can see, the pedal triangle does not have to be contained in Triangle ABC.
Problem 8

What if P is on one of the vertices of Triangle ABC?

The pedal triangle forms the altitude from the vertice to the corresponding base.
Problem 9

As we saw in Problem 8, when P is on one of the vertices, the pedal triangle becomes a straight line.

Thus, it becomes a degenerate triangle, called the Simson Line.

There are multiple conditions that can be met in order to creat the Simson Line.

Placing the Point P on any of the three vertices of Triangle ABC creates the line.

Placing Point P on the circumcircle of Triangle ABC will also create the Simson Line.
Problem 10 and 11

Construct the circumcircle of Triangle ABC. Now create a larger circle around the same center point C.

Animate Point P around this circle while tracing the midpoints of Triangle RST. What do you see?

Here is the construction you get for Problem 10. Click here to see Point P animated

around the circle and the resultant traces of the midpoints. You may click the "Move Point" button

to move the path of Point P onto the circumcircle of Triangle ABC.

It is interesting to note that the traces create ellipses distinct from each other. These ellipses

originally move around the vertices of Triangle ABC, but as the path of Point P converges onto

the circumcenter, the ellipses pass through each vertex.
Problem 11a

Construct lines on the sides of the Pedal Triangle and trace them as P moves around in a circular path.

Notice that the traces look to create a triangular figure similar to Triangle ABC.
Problem 11b

Moving Point P onto the circumcircle creates a similar but smaller figure.

Note that this illustrates the envelope of the Simson Line as described in Problem 9.
Problem 12

Repeat this process with the P travelling around a circle centered on C

but with a smaller radius than the circumcircle.

Problem 13

Is there a point on the circumcircle for P that has side AC as its Simson line? AB? BC?

For this solution, I created diameter lines of the circumcircle from each vertex.

I labeled these points A', B', and C'.

Click here to move Point P to each of these new points.

As you can see, when P is on one of the prime points, the pedal triangle

has one of the sides of Triangle ABC as its Simson Line.
Problem 14

Now, connect Point P on the circumcircle to the orthcocenter of Triangle ABC. Like so:

How do the Simson Line and Segment PH intersect? Click here to watch move P around the circle.

As you can see, the two segments overlap when P is on the vertices of Triangle ABC.

This is to be expected since we saw that the Simson Line was an altitude of Triangle ABC in Problem 8.
Problem 15

Select two pedal points on the circumcircle and construct their Simson lines.

Compare the angle of intersection of the two Simson lines with the angular measure

of the arc between the two pedal points.

Here is my construction:

The new Pedal Point is labeled P2, and was created from Triangle A2, B2, C2.

The Pedal Triangle, currently in the form of Simson Line, are formed by the points R2, S2, T2.

Problem 16

Animate the Pedal point P about the incircle of ABC.

Trace the loci of the midpoints of the sides. What curves result?

First, here is the construction:

After one revolution, the traces will look like this:

Now, repeat if ABC is a right triangle. Here is what it looks like:

Notice that one of the midpoint traces creates a circle tangent to AC and BC.
Problem 17

Construct an excircle of triangle ABC. Animate the Pedal point P about the excircle

and trace the loci of the midpoints of the sides of the Pedal triangle. What curves result?

Look at the angle bisectors through the center of the excircle. How are the loci positioned

with respect to the angle bisectors?