**Problem. **Show that a triangle is a right triangle if and only if

.

Vietnam 1981

**Solution.** (By removeablesingularity)

Suppose that is a a right triangle. Then . Check that

,

and hence the equation will be satisfied.

Now, suppose that the equation is satisfied. If the left-hand-side (LHS) is squared, we obtain

and if the right-hand-side (RHS) is squared, we obtain

.

Recall the trigonometric identities:

- (for instance, )

The equation above after squared then can rewritten as

.

Let us write . Then,

which factorizes to

Hence, or or which consecutively implies , , or . Hence, if the equation is satisfied, triangle has to be a right triangle.

Squaring both sides and a manipulation gives

writing yield

or .

This gives or .

Mantap Kak. Terima kasih hehe.