How To Draw A Circle Inside A Triangle

Incircle of a Triangle

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.

Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Copy a line segment
Sum of n line segments
Difference of two line segments
Perpendicular bisector of a line segment
Perpendicular from a line at a point
Perpendicular from a line through a point
Perpendicular from endpoint of a ray
Divide a segment into n equal parts
Parallel line through a point (angle copy)
Parallel line through a point (rhombus)
Parallel line through a point (translation)

Angles

Bisecting an angle
Copy an angle
Construct a 30° angle
Construct a 45° angle
Construct a 60° angle
Construct a 90° angle (right angle)
Sum of n angles
Difference of two angles
Supplementary angle
Complementary angle
Constructing 75° 105° 120° 135° 150° angles and more

Triangles

Copy a triangle
Isosceles triangle, given base and side
Isosceles triangle, given base and altitude
Isosceles triangle, given leg and apex angle
Equilateral triangle
30-60-90 triangle, given the hypotenuse
Triangle, given 3 sides (sss)
Triangle, given one side and adjacent angles (asa)
Triangle, given two angles and non-included side (aas)
Triangle, given two sides and included angle (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Right Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Right Triangle, given hypotenuse and one angle (HA)
Right Triangle, given one leg and one angle (LA)

Triangle Centers

Triangle incenter
Triangle circumcenter
Triangle orthocenter
Triangle centroid

Circles, Arcs and Ellipses

Finding the center of a circle
Circle given 3 points
Tangent at a point on the circle
Tangents through an external point
Tangents to two circles (external)
Tangents to two circles (internal)
Incircle of a triangle
Focus points of a given ellipse
Circumcircle of a triangle

Polygons

Square given one side
Square inscribed in a circle
Hexagon given one side
Hexagon inscribed in a given circle
Pentagon inscribed in a given circle

Non-Euclidean constructions

Construct an ellipse with string and pins
Find the center of a circle with any right-angled object