Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 20.

In this exploration, you will you will learn discover a special relationship about one of the special triangle centers. Follow the steps and keep those geometric senses tuned.

- Draw a circle and create three points (A, B, & C, above) on it to form a triangle.
- Create a triangle connecting the three points.
- Now create one more point on the circle (P in my image).
- Select Side AB of the triangle and use the transform menu to make it a mirror line. Then select point P and reflect it over the line.
- Repeat this process to reflect Point P over sides BC and AC.
- Now connect the three reflection points and note that they form a line.
- Move Point P around and observe the line. Can you find a stationary point?
- Highlight all three of the reflected points and Use the "TRACE" command under the
**DISPLAY**menu to make each point traceable. - Now, select P and move it around the center. What do you observe about the traced paths of the reflected points. Can you see a point the line always passes through? Can you figure out what it is? If you really can't find it, scroll down a little and we'll put a clue for you, since you need this for the last part of the exploration.
- Open a new page and create the point you found in part 9). What happens when you reflect it over each side of the triangle?
- Did you see some circles? Where do they intersect? Where are their centers?
- What other things have you observed in this exploration?

HINT: Try drawing some altitudes to the triangle.