Construction: Incenter

Drawing diagrams : Constructing the incenter and incircle.

What I usually do :

Instead of drawing the triangle first, I draw the circle first. Then, it is easy to construct the triangle.

However, what if you need to construct multiple incircles? Then, the above only works on one of the incircles.

Well then,

Step 1 : Draw the triangle, and its angle bisectors. (which you can estimate according to the size of the angle)

Step 2 : The intersection is the center you need.

Step 3 : To draw the incircle, drop a perpendicular from the incenter to it's sides. Then, use a compass with center at the incenter and the pencil at any perpendicular you marked.

The excircle can be dealt similarly. (But draw the circle first if possible)

The incenter of a triangle is equidistant from all three sides of the triangle.

We can think about it this way: Dropping an altitude to both of the sides, we get two congruent triangles (and since corresponding points of congruent triangles are congruent, the distance to the sides are congruent.

Now, let's draw an angle bisector of a triangle ABC

So that line is equidistant from AB and AC.

Now what? Think about it before going on...

Draw another angle bisector! If every point on the first angle bisector is equidistant from AB and AC, and every point on the second one is equidistant from AC and BC, then the point on both angle bisectors (the intersection point) is equidistant from all three sides!

So, we have successfully constructed the incenter of ABC.