STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 4

4. Given a triangle, draw a circle inside it which touches all three sides.

Solution

1. The trick here is to notice that each side of the triangle will be a tangent to the inscribed circle.
2. Pick any one point of the triangle, a. Let the sides of the triangle originating from a be B and C.
3. Construct the bisector of B and C according to the previous question that passes through the third side of the triangle. Let this bisector be D.
4. Repeat this procedure with each of the other corners of the triangle. The bisectors will all meet at a point inside the triangle, e, which is the centre of the inscribed circle.
5. It now remains to set the compasses to the appropriate radius. To do this, we drop a normal from e to one of the sides of the triangle (say, B) as follows: firstly, extend the side B into the line B'.
6. Draw a circle, centre e, radius ea.
7. Let the two points of intersection of this circle with B' be f and g.
8. Find the midpoint of f and g as described in the answer to question 1. Call this h.
9. the line eh will be normal (at right angles) to the line B. Draw a circle, centred on e, of radius eh. This is the required inscribed circle.