ABC is a triangle with, as usual, AB = c, CA = b. Find necessary and sufficient conditions for b^{2}c^{2}/(2bc cos A) = b^{2} + c^{2} - 2bc cos A.

**Solution**

Answer: BC = AB or CA.

We have that a^{2} = b^{2} + c^{2} - 2bc cos A. So the condition implies that (b^{2} + c^{2} - a^{2})/(2bc) = cos A = bc/(2a^{2}). Hence, b^{2}c^{2} = a^{2}(b^{2} + c^{2} - a^{2}), so (a^{2} - b^{2})(a^{2} - c^{2}) = 0. Hence a = b or c.

Conversely, if a = b, then the altitude from C meets AB at its midpoint and so cos A = cos B = c/(2a) = bc/(2a^{2}). Hence a^{2} = bc/(2 cos A), so (b^{2} + c^{2} - 2bc cos A) = b^{2}c^{2}/(2bc cos A).

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002