ABC is a triangle with, as usual, AB = c, CA = b. Find necessary and sufficient conditions for b2c2/(2bc cos A) = b2 + c2 - 2bc cos A.
Answer: BC = AB or CA.
We have that a2 = b2 + c2 - 2bc cos A. So the condition implies that (b2 + c2 - a2)/(2bc) = cos A = bc/(2a2). Hence, b2c2 = a2(b2 + c2 - a2), so (a2 - b2)(a2 - c2) = 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/(2a2). Hence a2 = bc/(2 cos A), so (b2 + c2 - 2bc cos A) = b2c2/(2bc cos A).
12th Putnam 1952
© John Scholes
5 Mar 2002