Sue Liu, S5, Madras College sent in a good solution which shows that if
A, B and C are angles in a triangle and
tan(A - B) + tan(B - C) + tan(C - A) = 0
then the triangle is isosceles. Can you prove a stronger result?
We start with the expression
tan(A - B) + tan(B - C) + tan(C - A) = 0.
Write X = A - C and Y = B - C, then the given expression becomes
tan(X - Y) + tanY + tan-X = 0.
This gives
tan(X - Y) = tanX - tanY
and we know the identity
tan(X - Y) =
tanX - tanY1 - tanX tanY
.
Hence either
tanX = tanY (1)
or
tanX tanY = 0 (2)
In case (1) we show that the angles X and Y are equal.
|X - Y| = |A - B| < A + B < 180 °
and the tan function is periodic with period 180 degrees so
X = Y. This gives A - C = B - C hence A = B, so the triangle is
isosceles.
In case (2), either tanX = 0 or tanY = 0, hence A = C or B = C and in all the cases the triangle is isosceles.