How to Find the Incenter of a Triangle
Theory
Incircle, Incenter and Angle Bisector
The angle bisectors of the angles in a triangle have one common point of intersection. This point is called the incenter, and it is in the center of the circle that just barely fits inside the triangle. The circle touches all the sides of the triangle—the triangle’s sides are all tangent to the circle. This circle is called the inscribed circle, or incircle.
If you call the triangle’s area , you get the following formula for the radius :
where , and are the lengths of each side in the triangle.
When you are asked to find the inscribed circle of a triangle, you have to bisect two of the angles belonging to that triangle. At the point of intersection between the angle bisectors, you find the incenter. Put the needle point of your compass on the incenter and construct the circle that barely touches each of the sides of the triangle.
Example 1
A triangle has the sides , and . Construct the incircle of the triangle.
Before you construct the incircle, you need to construct the triangle with the given measurements. Start with the line . Set the compass’s radius to 3 and make a faint circle with the center . Then, set the compass’s radius to 6 and make a faint circle with the center . The corner appears as the intersection between the two circles. Then you end up with the following triangle:
Then you construct the angle bisector for two of the sides. At the intersection between these, you have the incenter, which you call .
Construct a normal from the incenter down to one of the sides of the triangle. You can call the point of intersection between the normal and its belonging side . Then you construct a circle with center and radius . This circle is now tangent to all the sides of your triangle.
