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.