1967 IMO Problems/Problem 4
Let and
be any two acute-angled triangles. Consider all triangles
that are similar to
(so that vertices
,
,
correspond to vertices
,
,
, respectively) and circumscribed about triangle
(where
lies on
,
on
, and
on
). Of all such possible triangles, determine the one with maximum area, and construct it.
Solution
We construct a point inside
s.t.
, where
are a permutation of
. Now construct the three circles
. We obtain any of the triangles
circumscribed to
and similar to
by selecting
on
, then taking
, and then
(a quick angle chase shows that
are also colinear).
We now want to maximize . Clearly,
always has the same shape (i.e. all triangles
are similar), so we actually want to maximize
. This happens when
is the diameter of
. Then
, so
will also be the diameter of
. In the same way we show that
is the diameter of
, so everything is maximized, as we wanted.
This solution was posted and copyrighted by grobber. The thread can be found here: [1]
Solution 2
Since all the triangles circumscribed to
are similar, the one with maximum area will be the one with maximum sides, or
equivalently, the one with maximum side
. So we will try to maximize
.
(Solution by pf02, September 2024)
TO BE CONTINUED. I AM SAVING MID WAY SO AS NOT TO LOSE WORK DONE SO FAR.
See Also
1967 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |