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Incircle of a Triangle
Instructions Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.
(If there is no image below, see support page.)
Explanation of method
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.
The first six steps establish the incenter and are identical to those in Constructing the Incenter of a Triangle
Step-by-step Instructions
| Step 1 |
Place the compass point on any of the triangle's vertices.
Adjust the compass to a medium width setting. The exact width is not important. |
| Step 2 |
Without changing the compass width, strike an arc across each adjacent side. |
| Step 3 |
Change the compass 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 compass width for each. |
| Step 4 |
Using the straightedge, draw a line from the vertex of the triangle to where the last two arcs cross. |
| Step 5 |
Repeat all of the above at any other vertex of the triangle. You will now have two new lines drawn. |
| Step 6 |
Where the two new lines intersect, mark a point as the incenter of the triangle. |
| Step 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.
( Recall that the incircle just touches each side. Each side is a
tangent
to the circle, so point M will be the point of contact for this tangent. )
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| Step 8 |
Place the compass on the incenter and set the width to point M.
This is the radius of the incircle, sometimes called the inradius of the triangle. |
| Step 9 |
Draw a full circle. |
| Step 10 |
Done. This is the incircle of the triangle |
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
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Non-Euclidean constructions
(C) 2007 Copyright John Page
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