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Constructing the incircle of a triangle

This is the step-by-step, printable version. If you PRINT this page, any ads will not be printed.

See also the animated version.

After doing this Your work should look like this
We start with the given triangle. Geometry construction with compass and straightedge or ruler or ruler
The steps 1-6 establish the incenter and are identical to those in Constructing the Incenter of a Triangle
1.  Place the compasses' point on any of the triangle's vertices. Adjust the compasses to a medium width setting. The exact width is not important. Geometry construction with compass and straightedge or ruler or ruler
2.  Without changing the compasses' width, strike an arc across each adjacent side. Geometry construction with compass and straightedge or ruler or ruler
3.  Change the compasses' width if desired, then from the point where each arc crosses the side, draw two arcs inside the triangle so that they cross each other, using the same compasses' width for each. Geometry construction with compass and straightedge or ruler or ruler
4.  Using the straightedge, draw a line from the vertex of the triangle to where the last two arcs cross. Geometry construction with compass and straightedge or ruler or ruler
5.  Repeat all of the above at any other vertex of the triangle. You will now have two new lines drawn. Geometry construction with compass and straightedge or ruler or ruler
6.  Where the two new lines intersect, mark a point as the incenter of the triangle. Geometry construction with compass and straightedge or ruler or ruler
Optional Step  Repeat steps 1-4 for the third vertex. This will convince you that the three angle bisectors do, in fact, always intersect at a single point. But two are enough to find that point.
7.  Draw the perpendicular from the incenter to a side of the triangle. Label the point where it meets the side M.

See Constructing a Perpendicular from a Point for this procedure.
Geometry construction with compass and straightedge or ruler or ruler
8.  Place the compasses on the incenter and set the width to point M. This is the radius of the incircle, sometimes called the inradius of the triangle. Geometry construction with compass and straightedge or ruler or ruler
9.  Draw a full circle. Geometry construction with compass and straightedge or ruler or ruler
10.  Done. This is the incircle of the triangle  

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions