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Incircle of a Triangle
Geometry construction using a compass and straightedge

As can be seen in Incenter of a Triangle, the three angle bisectors of any triangle always pass through its incenter. In this construction, we only use two, as this is sufficient to define the point where they intersect. We bisect the two angles using the method described in Bisecting an Angle. The point where the bisectors cross is the incenter. We then draw a circle that just touches the triangles's sides.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

The image below is the final drawing from the above animation.

  Argument Reason
1 I is the incenter of the triangle ABC. By construction.
See Triangle incenter construction for method and proof.
2 IM is perpendicular to AB By construction.
See Constructing a perpendicular to a line from a point for method and proof.
3 IM is the radius of the incircle From (2), M is the point of tangency
4 Circle center I is the incircle of the triangle Circle touching all three sides.

  - Q.E.D

Try it yourself

Click here for a printable incircle worksheet containing two problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
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Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions