
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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.
Constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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