1966 IMO Problems/Problem 2
Let ,
, and
be the lengths of the sides of a triangle, and
respectively, the angles opposite these sides. If,
Prove that the triangle is isosceles.
Solution
We'll prove that the triangle is isosceles with .
We'll prove that
. Assume by way of contradiction WLOG that
.
First notice that as
then and the identity
our equation becomes:
Using the identity
and inserting this into the above equation we get:
Now, since
and the definitions of
being part of the definition of a triangle,
.
Now,
(as
and the angles are positive),
, and furthermore,
. By all the above,
Which contradicts our assumption, thus
. By the symmetry of the condition, using the same arguments,
. Hence
.
See Also
1966 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |