# Different Circles and Centers Connected with Triangle – Part2

## Different Circles and Centers Connected with Triangle – Part2

Incircle and Incenter:

The point of intersection of the internal bisectors of a triangle is called the incenter of the triangle. Also, it is the center of the circle touching all the three sides internally. Incenter always lies inside the triangle.

Point AFIE, BDIF, and CEID are concyclic

Internal bisector AP divided side BC in the ratio AB:AC

$$\frac{BP}{PC}=\frac{AB}{AC}=\frac{c}{b}$$,

$$\Rightarrow BP=ck,\ CP=bk$$,

BP + CP = a

ck + bk = a

k = a/(b+c)

BP = ac/(b+c) and

CP = ab/(b+c)

Similarly

AQ = cb/(a+c) and

CQ = ab/(a+c)

And AR = bc/(a+b), BR = ac/(a+b)

$$r=\left( s-a \right)\tan \frac{A}{2}=\left( s-b \right)\tan \frac{B}{2}=\left( s-c \right)\tan \frac{C}{2}$$,

Proof: $$\left( s-a \right)\tan \frac{A}{2}=\left( s-a \right)\sqrt{\frac{\left( s-b \right)\left( s-c \right)}{s\left( s-a \right)}}$$,

$$\left( \because \tan \frac{A}{2}=\sqrt{\frac{\left( s-b \right)\left( s-c \right)}{s\left( s-a \right)}} \right)$$,

$$\left( s-a \right)\tan \frac{A}{2}=\sqrt{\frac{{{\left( s-a \right)}^{2}}\left( s-b \right)\left( s-c \right)}{s\left( s-a \right)}}$$,

$$=\sqrt{\frac{\left( s-a \right)\left( s-b \right)\left( s-c \right)}{s}}$$,

$$=\sqrt{\frac{\left( s-a \right)\left( s-b \right)\left( s-c \right)}{s}}\times \frac{\sqrt{s}}{\sqrt{s}}$$,

$$=\sqrt{\frac{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}{{{s}^{2}}}}$$,

$$=\frac{1}{s}\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$$,

$$\left( \because \Delta =\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} \right)$$,

$$r=\frac{\Delta }{s}$$,

Similarly, r = $$\left( s-b \right)\tan \frac{B}{2}=\frac{\Delta }{s}$$ and r = $$\left( s-c \right)\tan \frac{C}{2}=\frac{\Delta }{s}$$.