Difference between revisions of "1993 USAMO Problems/Problem 2"
(→Resources) |
|||
Line 65: | Line 65: | ||
{{USAMO box|year=1993|num-b=1|num-a=3}} | {{USAMO box|year=1993|num-b=1|num-a=3}} | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks] | ||
+ | {{MAA Notice}} |
Revision as of 13:29, 4 July 2013
Contents
Problem 2
Let be a convex quadrilateral such that diagonals
and
intersect at right angles, and let
be their intersection. Prove that the reflections of
across
,
,
,
are concyclic.
Solution
Diagram
![]() |
Work
Let ,
,
,
be the foot of the altitute from point
of
,
,
,
.
Note that reflection of over the 4 lines is
with a scale of
with center
. Thus, if
is cyclic, then the reflections are cyclic.
is right angle and so is
. Thus,
is cyclic with
being the diameter of the circumcircle.
Follow that, because they inscribe the same angle.
Similarly ,
,
.
Futhermore, .
Thus, and
are supplementary and follows that,
is cyclic.
Resources
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.