Explorations in Geometry
Draw a triangle and construct the angle bisectors from each vertex to the opposite side. Find the point where these angle bisectors intersect.
1) Can you change the shape of the triangle so that the three angle bisectors do NOT intersect at the same point? Can you make them intersect outside the triangle.
2) Measure the distance from the intersection of the angle bisectors to the sides of the triangle. What pattern do you notice.
3) For each of the angle bisectors measure the distance from the intersection of the bisectors to a vertex, then measure the distance from the intersection to the side opposite the vertex. Can you observe a pattern in the ratio of these lengths? (hint, you might want to find the lengths of the three sides) Work hard on this one. This theorem was discovered by Mr Ballew, so expect him to think it is VERY important.
4) Find the perimeter and the area of the triangle. Can you find a relationship between the distance you found in question 2 and these two values?
5) Explore the angle bisectors of triangles and see what else you can discover. Write a brief paragrah summarizing what you have found. Be sure to answer the specific questions above, and any other discoveries you have noted.
6) There is a circle that plays a part in this diagram, especially in parts 2 and 4. What circle is it? And where is it's center?