Incenter, incircle and angle bisectors in triangle/polygons

 Incenter:

1. Why are 3 angle bisectors concurrent?
Proof:
A. Angle bisector by definition is equidistant from the 2 adjacent sides. 
B. So the point where 2 angle bisectors meet is equidistant from all 3 sides.
C. Hence the 3rd angle bisector will also pass through that point.

2. Since a triangle will always have an incenter, it will always have an incircle.

3. Easy to prove that the radius of incenter r = A/s where A is area of triangle and s = perimeter/2.
Steps:
A. Since incenter is equidistant from all sides, let's drop perpendiculars to sides, its length is r.
B. All sides are tangents to the circle and hence meet those 3 perpendiculars.
C. Now 3 sub triangles, area of each is a*r/2 where a is the corresponding side length.
D. Add all these sub areas and equate to main area to get the proof.

4. Same formula applies to any polygon which has an incircle.

5. Not all polygons have an incenter though. They may or may not have it.

6. But all regular polygons have an incenter.


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