Triangles and Circles
A circumcenter of a triangle is the point where all three perpendicular bisectors of a the sides of a triangle meet. In the triangle below,
1. Create all three perpendicular bisectors.
2. Create a circle with the center at the circumcenter that passes through one of the vertices of the triangle.
Change the size of the triangle. What do you notice about the circle?
What does this tell you about the circumcenter of a triangle?
What determines if the circumcenter is inside the triangle, on the triangle, or outside the triangle?
A incenter of a triangle is the point where all three angle bisectors of a triangle intersect. In the triangle below,
1. Create all three angle bisectors.
2. Create a line through the incenter that is perpendicular to a side of the triangle. (Pick any side you want.)
3. Create a circle with the center at the incenter that passes through the point where the perpendicular line intersects the side of the triangle.
Change the shape of the triangle. What do you notice about the circle?
Will the circle always touch all three sides of the triangle? Why or why not?
Will the circle always be trapped inside the triangle? Why or why not?
CHALLENGE:
Create two circles below that meet the following requirements.
1. One of the circles must always touch all three vertices of the triangle even when the triangle changes shape.
2. The other circle must always touch all three sides of the triangle and never cross over any side of the triangle.