Definition of Incircle

The incircle of a triangle is a circle that is located in the interior of a triangle and is tangent to all three sides of the triangle.

Constructing an Incircle (interactively!)

To construct the incircle of a triangle, follow these steps:

• Step 1. Construct the angle bisectors of the triangle. (Since all three angle bisectors intersect at a common point, it is enough to construct any two angle bisectors.)
• Step 2. Mark the incenter, the intersection of the angle bisectors.
• Step 3. Drop a perpendicular from the incenter to one of the sides of the triangle and mark the point where it intersects the side.
• Step 4. Draw a circle that has center at the incenter and that passes through the foot of the perpendicular drawn in Step 3.
• You can view the steps in the following applet. Drag any of the vertices of the triangle to change its shape.

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A Closer Look at the Construction of the Incircle — Why Does It Work?

In the interactive math tool below, point I is the incenter of triangle ABC. Think about the following reasoning:

• Triangles API and AQI are congruent. Can you see why?
  
• The circle with radius IQ centered at I will pass through point P. Why?
  
• Why is this circle tangent to sides AB and AC?
  
• Will it also be tangent to side BC? Why or why not?

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Answer

Triangles API and AQI are congruent because they are both right triangles with the same hypotenuse and with congruent angles at vertex A. Segments IP and IQ are congruent since they are corresponding sides of congruent triangles API and AQI, so the circle centered at I going through point Q will pass through point P. Segments IP and IQ are perpendicular to sides AC and AB, respectively, so the circle centered at I is tangent to those two sides. It will also be tangent to side BC using a similar argument. Use the button to show the incircle.

Back to incenter

Back to angle bisector

Back to Overview of Lines

References and Credits