Difference between revisions of "2017 USAJMO Problems/Problem 3"
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+ | <cmath>A = (1, 0, 0)</cmath> | ||
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+ | <cmath>C = (0, 0, 1)</cmath> | ||
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Revision as of 21:59, 5 May 2017
Contents
[hide]Problem
() Let
be an equilateral triangle and let
be a point on its circumcircle. Let lines
and
intersect at
; let lines
and
intersect at
; and let lines
and
intersect at
. Prove that the area of triangle
is twice that of triangle
.
Solution 1
WLOG, let . Let
, and
. After some angle chasing, we find that
and
. Therefore,
~
.
Lemma 1: If , then
.
This lemma results directly from the fact that
~
;
, or
.
Lemma 2: .
We see that
, as desired.
Lemma 3: .
We see that
However, after some angle chasing and by the Law of Sines in
, we have
, or
, which implies the result.
By the area lemma, we have and
.
We see that . Thus, it suffices to show that
, or
. Rearranging, we find this to be equivalent to
, which is Lemma 3, so the result has been proven.
Solution 2
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2017 USAJMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |