Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 8

- Construct a triangle ABC and the angle Bisector of the angle at B.
- Measure the sides of the triangle and the two pieces formed on side AC by the angle bisector. Can you find a relationship between the measurements. Is it true for all shapes of triangles?
- Now measure the length of the angle biscetor segment itself. Can you find a relationship between this length and the measurements above? If you can, you've discovered a special case of Stewart's Thm.
- Find the area of the two sub-triangles ABD and BCD. Can you see a relationship between their sizes.
- Measure the perpendicular distance from points C and D to the line AB. What do you observe. Why might you have expected this relationship?
- Create the circle circumscribing ABC. Find the point where the angle bisector crosses the circle. Measure the length of the arc from this point to C and to A. What do you observe? Can you PROVE that it is always true?
- What other geometric or algebraic relationships can you find in this figure?
- Write a brief paragraph explaining what you have found in this exploration. Remember to answer all the specific questions above.