## Tuesday, October 21, 2008

### Elearn Geometry Problem 195

See complete Problem 195 at:

Area of a Triangle, Inradius, Exradii. Level: High School, SAT Prep, College geometry

1. lengths BC=a, CA=b, AB=c; s=(a+b+c)/2
S=sr
S=[EAB]+[EAC]-[EBC]=c(r_a)/2+b(r_a)/2-a(r_a)/2=(s-a)(r_a),=(s-b)(r_b)=(s-c)(r_c)
By Heron's formula:
S=√[s(s-a)(s-b)(s-c)]
S√[r(r_a)(r_b)(r_c)]=√[sr(s-a)(r_a)(s-b)(r_b)(s-c)(r_c)]=S²
S=√[r(r_a)(r_b)(r_c)]

2. let BC = a, AB = c and AC = b and s=(a+b+c)/2
use the formulae, ra = S/(s-a), rb = S/(s-b) and rc = S/(s-c) and finally r = S/s where S denotes the area of the triangle ABC.
now r*ra*rb*rc = (S*S*S*S)/s(s-a)(s-b)(s-c)
but s(s-a)(s-b)(s-c) = S^2 and this follows from heron's formula.
which implies that r*ra*rb*rc = S^4/S^2 = S^2.
so that S = (r*ra*rb*rc)^(1/2).
Q. E. D.

3. To Antonio: When I try to get the comments or solutions of problem 196, I only get those of problem 195. Is there any bug in the links?

1. To Nilton (problem 196), thanks for your information, the link has been corrected.

4. Do you have similar blog/site for other maths topics? Your blog is excellent.