Difference between revisions of "2014 USAJMO Problems/Problem 2"
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Note: It's easy to show that for any point <math>H</math> on <math>\overline{PQ}</math> except the midpoint, Points B and C can be validly defined to make an acute, non-equilateral triangle. | Note: It's easy to show that for any point <math>H</math> on <math>\overline{PQ}</math> except the midpoint, Points B and C can be validly defined to make an acute, non-equilateral triangle. | ||
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+ | ==See Also== | ||
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+ | {{USAJMO box|year=2014|num-b=1|num-a=3}} |
Revision as of 10:38, 10 March 2015
Problem
Let be a non-equilateral, acute triangle with
, and let
and
denote the circumcenter and orthocenter of
, respectively.
(a) Prove that line intersects both segments
and
.
(b) Line intersects segments
and
at
and
, respectively. Denote by
and
the respective areas of triangle
and quadrilateral
. Determine the range of possible values for
.
Solution
Claim: is the reflection of
over the angle bisector of
(henceforth 'the' reflection)
Proof: Let be the reflection of
, and let
be the reflection of
.
Then reflection takes to
.
is equilateral, and
lies on the perpendicular bisector of
It's well known that lies strictly inside
(since it's acute), meaning that
from which it follows that
. Similarly,
. Since
lies on two altitudes,
is the orthocenter, as desired.
So is perpendicular to the angle bisector of
, which is the same line as the angle bisector of
, meaning that
is equilateral.
Let its side length be , and let
, where
because
lies strictly within
, as must
, the reflection of
. Also, it's easy to show that if
in a general triangle, it's equilateral, and we
is not equilateral. Hence H is not on the bisector of
. Let
intersect
at
.
Since and
are 30-60-90 triangles,
Similarly,
The ratio is
The denominator equals
where
can equal any value in
except
. Therefore, the denominator can equal any value in
, and the ratio is any value in
Note: It's easy to show that for any point on
except the midpoint, Points B and C can be validly defined to make an acute, non-equilateral triangle.
See Also
2014 USAJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |