Question

Three circles of radius a, b, c touch each other externally. The area of the triangle formed by joining their centres is:

(a) \(\sqrt{(a+b+c)abc}\)

(b) (a + b + c) \(\sqrt{ab+bc+ca}\)

(c) ab + bc + ca 

(d) None of these

Answer 1

(a) \(\sqrt{(a+b+c).a.b.c}\)

As shown in the figure, AB = a + b, BC = b + c, CA = a + c

∴ Area of ΔABC = \(\sqrt{s(s-AB)(s-BC)(s-CA)}\)

where, s = \(\frac{1}{2}\) (AB + BC + CA)

= \(\frac{a+b+b+c+c+a}{2}\) 

= a + b + c

∴ Area of ΔABC

=\(\sqrt{(a+b+c)[(a+b+c)-(a+b)][(a+b+c)-(b+c)][(a+b+c)-(c+a)]}\)

= \(\sqrt{(a+b+c).a.b.c}\)