Area of Triangle in Terms of Inradius 증명

Theorem
Let △ABC be a triangle whose sides are of lengths a,b,c .

Then the area A of △ABC is given by:

A=rs

where:

r is the inradius of △ABC

s=a+b+c 2 is the semiperimeter of △ABC .

Proof

Let I be the incenter of △ABC .
Let r be the inradius of △ABC .
The total area of △ABC is equal to the sum of the areas of the triangle formed by the vertices of △ABC and its incenter:

A=Area(△AIB)+Area(△BIC)+Area(△CIA)

Let AB , BC and CA be the bases of △AIB,△BIC,△CIA respectively.
The lengths of AB , BC and CA respectively are c,a,b .
The altitude of each of these triangles is r .
Thus from Area of Triangle in Terms of Side and Altitude:

				Area(△AIB)	=		cr 2
				Area(△BIC)	=		ar 2
				Area(△CIA)	=		br 2

Thus:

A=ra+b+c 2

That is:

A=rs

where s=a+b+c 2 is the semiperimeter of △ABC .
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