CK12 - Geometry

1. Consider.


2. We can constructthe anglebisectorsof and intersectingat point as follows.


3. We will showthat point is equidistantfrom sides , , and and that is on the bisector

of.

4. Constructperpendicularline segmentsfrom point to sides , , and as follows:


5. Since is on the bisectorsof and , then by Theorem5-5,.

Therefore, is equidistantfrom sides , , and.

6. Since is equidistantfrom and , Theorem5-6 appliesand we musthavethat is on the

anglebisectorof.

The point has a specialproperty. Sinceit is equidistantfrom eachside of the triangle,we can see that

is the centerof a circlethat lies withinthe triangle.We say that the circleisinscribedwithinthe triangle

and the point is calledtheincenterof the triangle.This is illustratedin the followingfigure.

Example 1