Incircles and Excircles
Contents
Incircles and Incenters
Introduction
How would you draw a circle inside a triangle, touching all three sides? It is actually not too complex. Simply bisect each of the angles of the triangle; the point where they meet is the center of the circle! Then use a compass to draw the circle. But what else did you discover doing this?
- The three angle bisectors all meet at one point.
- This point is equidistant from all three sides.
In order to prove these statements and to explore further, we establish some notation.
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Let , and be the angle bisectors.
The incenter is the point where the angle bisectors meet.Let and be the perpendiculars from the incenter to each of the sides.
The incircle is the inscribed circle of the triangle that touches all three sides.The inradius is the radius of the incircle.
Now we prove the statements discovered in the introduction.
In a triangle , the angle bisectors of the three angles are concurrent at the incenter . Also, the incenter is the center of the incircle inscribed in the triangle.
Given place point on such that bisects and place point on such that bisects Let be their point of intersection. Then place point on such that place point on such that and place point on such that Finally, place point on such that passes through point
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and have the following congruences:
- because they are both right angles.
- because is the angle bisector.
- because of the reflexive property of congruence.
Thus, by AAS, In a similar fashion, it can be proven that Then, by CPCTC (congruent parts of congruent triangles are congruent) and the transitive property of congruence,
Now and have the following congruences:
- as stated earlier.
- because they are both right angles.
- because of the reflexive property of congruence.
Thus, by HL (hypotenuse-leg theorem), By CPCTC, Hence, is the angle bisector of and all three angle bisectors meet at point
Since there exists a circle centered at that passes through and Furthermore, since these segments are perpendicular to the sides of the triangle, the circle is internally tangent to the triangle at each of these points. Hence, the incenter is located at point
Excircles and Excenters
If we extend two of the sides of the triangle, we can get a similar configuration.
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Note that these notations cycle for all three ways to extend two sides is the excenter opposite . It has two main properties:
- The angle bisectors of are all concurrent at .
- is the center of the excircle which is the circle tangent to and to the extensions of and .
is the radius of the excircle.
The proofs of these results are very similar to those with incircles, so they are left to the reader.
Main Properties and Examples
There are many amazing properties of these configurations, but here are the main ones. In these theorems the semi-perimeter , and the area of a triangle is denoted .
Elementary Length Formulae:
First we prove two similar theorems related to lengths.
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Tangents from the same point are equal, so (and cyclic results). Then it follows that , but , so
and the result follows immediately.
The argument is very similar for the other two results, so it is left to the reader.
The proof of this theorem is quite similar and is left to the reader.
Area Formulae:
This is a beautiful theorem about areas:
The proof is left to the reader for now.
More Advanced Useful Properties
These more advanced, but useful properties will be listed for the reader to prove (as exercises).
Radii Relationships:
These are very useful when dealing with problems involving the inradius and the exradii. Let be the circumradius.
And here is one of my favorites:
Computing Lengths: