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The Incenter and Circumcenter of a Triangle

The Incenter of a Triangle

When a circle and a line meet at a single point, the line is said to tangent to the circle. The point of contact between the circle and the tangent line is called the point of tangency, and the tangent is perpendicular to the radius drawn to the point of tangency.

Properties of the Incenter:

The angle bisectors of the three angles of a triangle meet at a single point, called the incenter.
The distances from the incenter to the three sides of the triangle are equal.

[Explanation]
In

, let the intersection of the angle bisectors of

and

be point

. From

, drop perpendiculars to the sides

, and

, meeting them at points

, and

, respectively.

Since lies on the bisectors of and ,

Connecting to , consider and :

By congruence of

, it follows that

, and hence

bisects

. Therefore, the angle bisectors of

meet at a single point

, the incenter.

Furthermore, since

, a circle centered at

with radius

will touch each side of

at points

, and

. Such a circle is called the incircle of

, and its center

is the incenter.

The Circumcenter of a Triangle

Properties of the Circumcenter:

The perpendicular bisectors of the three sides of a triangle meet at a single point, called the circumcenter.
The distances from the circumcenter to the three vertices of the triangle are equal.

[Explanation]
In

, let the perpendicular bisectors of

and

meet at point

Since lies on the perpendicular bisectors of and :

Let be the foot of the perpendicular from to . Consider and :

By congruence of

, it follows that

, and hence

is the perpendicular bisector of

. Therefore, the perpendicular bisectors of

meet at a single point

, the circumcenter.

Furthermore, since

, a circle centered at

with radius

will pass through all three vertices of

. Such a circle is called the circumcircle of

, and its center

is the circumcenter.