Difference between revisions of "1967 IMO Problems/Problem 4"
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This solution was posted and copyrighted by grobber. The thread can be found here: [https://aops.com/community/p139266] | This solution was posted and copyrighted by grobber. The thread can be found here: [https://aops.com/community/p139266] | ||
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+ | ==Solution 2== | ||
+ | |||
+ | Since all the triangles <math>\triangle ABC</math> circumscribed to <math>\triangle A_0B_0C_0</math> | ||
+ | are similar, the one with maximum area will be the one with maximum sides, or | ||
+ | equivalently, the one with maximum side <math>BC</math>. So we will try to maximize <math>BC</math>. | ||
+ | |||
+ | [[File:Prob_1967_4_fig1.png|800px]] | ||
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+ | |||
+ | |||
+ | (Solution by pf02, September 2024) | ||
+ | |||
+ | TO BE CONTINUED. I AM SAVING MID WAY SO AS NOT TO LOSE WORK DONE SO FAR. | ||
== See Also == | == See Also == |
Revision as of 17:04, 3 September 2024
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 |