Problem
Prove that for a given triangle, there exists a circle which inscribes it.
Definitions
What is a Circle? A set of points equidistant from a center. What is a Triangle? A closed shape with three vertices and three faces. What does it mean for a Triangle to be inscribed in a Circle? The vertices of the Triangle lie on the circumference of the Circle.
Relevant Concepts
If we know one or more point/s on the circumference of the circle and we somehow figure out the center, we would get the radius for free(the distance between the point on the circumference and the center).
The center is equidistant from the circumference. So, we can solve the problem by proving that there exists a point such that it is equidistant from the three vertices of the triangle.
Walkthrough
When I talk about a point P being equidistant form two other points A and B. What do you think P is? The point in the middle of the A and B. Visualize the position of these three points. Now try to move P keeping it equidistant from A and B. You will discover that P is equidistant form A and B as long as it stays on a specific line. What can you say about that line?
Now do this for each possible pairs of vertices of the triangle i.e. vertex1 and vertex2, vertex1 and vertex3, vertex2 and vertex3. This gives us three lines. These three lines intersect at a point. This point is equidistant from the three vertices of the triangle. We have just found the center of the circle!
Exercise for the Reader
Do we really need three points for this process? Will two not suffice?
Given a Triangle, how many distinct Circles can it be inscribed in?
Given a Circle, how many distinct Triangles can be inscribed in it?
Can this process be done for all possible triangles?
Summary
The intersection of the perpendicular bisectors of the sides of a triangle gives the center of the circle in which it can be inscribed.
