1993 IMO Problems/Problem 2
Problem
Let be a point inside acute triangle
such that
and
.
(a) Calculate the ratio .
(b) Prove that the tangents at to the circumcircles of
and
are perpendicular.
Solution
Let us construct a point
satisfying the following conditions:
are on the same side of AC,
and
.
Hence .
Also considering directed angles mod ,
.
Also, .
.
Hence, .
Finally, we get .
For the second part, let the tangent to the circle be
and the tangent to the circle
be
.
due to the tangent-chord theorem.
for the same reason.
Hence,
We also have
.
which means circles
and
are orthogonal.
See Also
1993 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |