How Do You Construct the Incenter of a Triangle?

The incenter of a triangle is defined as the point where all three angle bisectors intersect. It can be found using a compass and a straight edge by constructing the angle bisectors of any two vertices of the triangle and marking the point where they cross. Two angle bisectors are sufficient to define the point where all three intersect, so the third angle bisector need not be drawn.

1. Make marks equidistant from the vertex on two adjacent legs of the triangle

Set the point of the compass on any vertex of the triangle, and set the compass distance so that it is less than the length of the shorter of the two adjacent legs. Make an arc on each of the legs without changing the setting, so that each arc is equidistant from the vertex.

2. Make intersecting arcs inside the triangle

Using the points where the arcs intersect the legs, construct two new arcs inside the triangle. Use the same compass distance for each new arc, and adjust the distance so that the two new arcs intersect.

3. Draw the angle bisector

Draw a line between the vertex and the point where the two arcs intersect using a straight edge. This line is the first angle bisector.

4. Repeat the process at another vertex

Chose another vertex and construct the angle bisector using the process described in steps 1 through 3. The point where the two angle bisectors intersect is the incenter of the triangle.

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