The exploration in this write-up deals with Pedal Triangle.

Click here to see construction of the Pedal Triangle. In this picture, the Pedal Triangle is the triangle in blue with point P being the pedal point. I have also included the orthocenter(H), the centroid(G), the incenter(I), and the circumcenter(C). In our exploration we were asked to use these various points as our pedal point. The observations made from these explorations are as follows.

1. When the pedal point is the centroid(G), and triangle ABC is equilateral the ratio of the area of the pedal triangle to triangle ABC is 1/4. The ratio of the perimeter of the pedal triangle to triangle ABC is 1/2 when triangle ABC is equilateral.

2. When the pedal point is the incenter(I), the ratio of the area of the pedal triangle to triangle ABC is 1/4, but I only found this to be true when triangle ABC is equilateral. Also when triangle ABC is equilateral, the ratio of the perimeter of the pedal triangle to triangle ABC is 1/2.

3. When the pedal point is the orthocenter(H), I did not find any correlation between the ratio of the areas or the perimeter. But I did find that when the triangle ABC is a right triangle is when the Simson Line occurs. This is when the three vertices of the pedal triangle or colinear.

4. When the pedal point is the circumcenter(C), I found that the ratio of the area of the pedal triangle to triangle ABC to be 1/4 when triangle ABC is equilateral. I also found that when triangle ABC is equilateral the ratio of the perimeter of the pedal triangle to triangle ABC is 1/2.

5. There is a point on the circumcircle for P that has side AC, AB, and BC as its Simson Line.

Click here to see a construction.

If I were to try these investigations with my students the following steps is what would be asked of them.

In the following activity you will be asked to explore Pedal Triangles.

**Problem**: Let triangle ABC be
any triangle. In the following steps you will use the center of the nine
point circle of triangle ABC, a point on triangle ABC, and one of the vertices
as a pedal point.

**Grade**: 11th

**Class**: Honors Geometry

**Objective**:

1.To explore Pedal Triangles with the pedal point being the center of the nine point circle of triangle ABC.

2. To explore the relationship between the Pedal Triangle and Triangle ABC when the pedal point is on the side of the triangle.

3. To explore the relationship between the Pedal triangle and Triangle ABC when the pedal point is any one of the vertices of triangle ABC.

**Part 1**

Step I: Construct any triangle ABC.

Step II: Construct the nine point triangle for triangle ABC.

Step III. Explore these constructions for different types of triangles.

a.) Do you see any relationship between the ratio of the areas of triangle ABC and the pedal triangle.

b.) Do you see any relationships between the ratio of the pedal triangle and triangle ABC.

Step IV: Is there a point where there is a Simson Line that occurs? Note that a Simson Line is when the vertices of the pedal triangle are colinear.

**Part 2**

Step I: Construct any triangle ABC.

Step II: Construct a point P on any side of triangle ABC.

Step III: Construct the pedal triangle.

Step IV: Explore these constructions for different types of triangles.

a.) Do you see any relationship between the ratio of the areas of triangle ABC and the pedal triangle.

b.) Do you see any relationships between the ratio of the pedal triangle and triangle ABC.

Step V: Is there a point where there is a Simson Line that occurs? Note that a Simson Line is when the vertices of the pedal triangle are colinear.

**Part 3**

Step I: Construct triangle ABC.

Step II: Place your pedal point P on any one of the vertices of triangle ABC.

Step III: Explore these constructions for different types of triangles.

a.) Do you see any relationship between the ratio of the areas of triangle ABC and the pedal triangle.

b.) Do you see any relationships between the ratio of the pedal triangle and triangle ABC.

Step IV: Is there a point where there is a Simson Line that occurs? Note that a Simson Line is when the vertices of the pedal triangle are colinear.

**Part 4**

You must pick one of the following to do on top of the steps asked above.

1. Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

2. Follow the directions in number one, but use the circumcircle as the path.

a.) Construct lines (not segments) on the sides of the Pedal Triangle. Trace the lines as the Pedal Point is moved along different paths

b.) Find the envelope of the Simson line as the Pedal Point is moved along the circumcircle. NOTE: Trace the image of the line, not the segment.

3. Animate the Pedal Point P about the incircle of ABC. Trace the loci of the midpoints of the sides.

a.) What curves result?

b.) Repeat if ABC is a right triangle.

**Part 5**

Please organize all of your findings and construct a write-up of your findings and observations. Please make sure to include pictures and scripts of your work. Make sure to save them on your disk so that I can see them.

You will be graded on neatness, correct mathematical language, and grammar and spelling. And of course effort and creativity will be also taken into account.