TS EAMCET 2015 Solved Paper

Consider the standard notation followed in triangle geometry.
Let r be the radius of the incircle of the triangle.
Let r1,r2,r3 be the radii of the excircles drawn opposite to the vertices A,B,C respectively.
Let s be the semi-perimeter of the triangle where s=‌

a+b+c

2

Let a,b,c be the lengths of the sides of the triangles opposite to the vertices A,B,C respectively..
Let ∆ be the area of the triangle.
We know,

∆=r.s=r1(s−a)=r2(s−b)=r3(s−c)
∴‌

1

r

=‌

s

∆

‌

1

r1

=‌

s−a

∆

‌

1

r2

=‌

s−b

∆

‌‌‌

1

r3

=‌

s−c

∆

⇒‌

1

r12

+‌

1

r22

+‌

1

r32

+‌

1

r2

=‌

1

∆2

((s−a)2+(s−b)2+(s−c)2+s2)
=‌

1

∆2

(s2−2as+a2+s2−2bs+b2+s2−2cs+c2+s2)

=‌

1

∆2

(4s2−2s(a+b+c)+(a2+b2+c2))
=‌

1

∆2

(4s2−4s2+(a2+b2+c2))(∵a+b+c=2s)
∴‌

1

r12

+‌

1

r22

+‌

1

r32

+‌

1

r2

=‌

(a2+b2+c2)

∆2