STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 5

5. Given a triangle, draw a circle outside it that touches all three corners.

Solution

1. Despite the superficial similarity to the previous question, this uses a different idea completely: namely, that each side of the triangle is a chord of the circumscribing circle; the normal bisector of a chord passes through the centre of the circle.
2. Choose any side of the triangle, A. Find the midpoint of A using the solution to question 1. Call this a.
3. Find the line normal to A that passes through a, as per the solution to question 2.
4. Repeat this process for the other two sides.
5. The normal bisectors will all meet at a point (not necessarily inside the triangle). This point, c, is the centre of the circumscribing circle.
6. Draw a circle, radius cd for any point for the triangle d. This is the required circle.