Question

The triangle ABC (with side lengths a,b,c as usual) satisfies loga² = logb² + logc² - log(2bc cosA). What can you say about this triangle?

Answer 1

a² = b²c²/2bcosA

cosA = bc/2a² = (b² + c² - a²)/2bc

⇒ b²c² = a²b² + a²c² – a⁴

⇒ b²(c² – a²) – a²(c² – a²) = 0

⇒ (c² – a²)(b² – a²) = 0

⇒ either c = a or a = b

Hence triangle is isosceles.