Difference between revisions of "1994 USAMO Problems/Problem 3"
(→Solution) |
(Reconstructed from page template) |
||
Line 163: | Line 163: | ||
Borrowed from https://mks.mff.cuni.cz/kalva/usa/usoln/usol943.html | Borrowed from https://mks.mff.cuni.cz/kalva/usa/usoln/usol943.html | ||
+ | |||
+ | ==See Also== | ||
+ | {{USAMO box|year=1994|num-b=2|num-a=4}} | ||
+ | {{MAA Notice}} | ||
+ | [[Category:Olympiad Geometry Problems]] |
Latest revision as of 07:02, 19 July 2016
Problem
A convex hexagon is inscribed in a circle such that
and diagonals
, and
are concurrent. Let
be the intersection of
and
. Prove that
.
Solution
Let the diagonals ,
,
meet at
.
First, let's show that the triangles and
are similar.
because
,
,
and
all lie on the circle, and
.
because
, and
,
,
,
and
all lie on the circle. Then,
Therefore, and
are similar, so
.
Next, let's show that and
are similar.
because
,
,
and
all lie on the circle, and
.
because
,
,
and
all lie on the circle.
because
, and
,
,
,
and
all lie on the circle. Then,
Therefore, and
are similar, so
.
Lastly, let's show that and
are similar.
Because and
,
,
and
all lie on the circle,
is parallel to
. So,
and
are similar, and
.
Putting it all together, .
Borrowed from https://mks.mff.cuni.cz/kalva/usa/usoln/usol943.html
See Also
1994 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.