Explorations in Geometry

with the

Geometerís 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.

    1. Draw a circle and create three points (A, B, & C, above) on it to form a triangle.
    2. Create a triangle connecting the three points.
    3. Now create one more point on the circle (P in my image).
    4. 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.
    5. Repeat this process to reflect Point P over sides BC and AC.
    6. Now connect the three reflection points and note that they form a line.
    7. Move Point P around and observe the line. Can you find a stationary point?
    8. Highlight all three of the reflected points and Use the "TRACE" command under the DISPLAY menu to make each point traceable.
    9. 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.
    10. 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?
    11. Did you see some circles? Where do they intersect? Where are their centers?
    12. What other things have you observed in this exploration?

HINT: Try drawing some altitudes to the triangle.