1988 IMO Problems/Problem 5
Problem
In a right-angled triangle let
be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles
intersect the sides
at the points
respectively. If
and
dnote the areas of triangles
and
respectively, show that
Solution
Lemma: Through the incenter of
draw a line that meets the sides
and
at
and
, then:
Proof of the lemma:
Consider the general case:
is any point on side
and
is a line cutting AB, AM, AC at P, N, Q. Then:
If is the incentre then
,
and
. Plug them in we get:
Back to the problem
Let and
be the areas of
and
and
be the intersection of
and
. Thus apply our formula in the two triangles we get:
and
Cancel out the term
, we get:
So we conclude
.
Hence and
, thus
and
. Thus
. So the area ratio is:
This solution was posted and copyrighted by shobber. The original thread for this problem can be found here: [1]
See Also
1988 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |