Difference between revisions of "1993 USAMO Problems/Problem 2"
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Similarly <math>\angle EWZ\cong \angle EDC</math>, <math>\angle EYX\cong \angle EBA</math>, <math>\angle EYZ\cong \angle ECD</math>. | Similarly <math>\angle EWZ\cong \angle EDC</math>, <math>\angle EYX\cong \angle EBA</math>, <math>\angle EYZ\cong \angle ECD</math>. | ||
− | <br/>Futhermore, <math>m\angle XYZ+m\angle XWZ= m\angle EWX+m\angle EYX+m\angle EYZ+m\angle EWZ=360^\circ-m\angle CED-m\angle AEB=180^\circ</math>. | + | <br/>Futhermore, <math>m\angle XYZ+m\angle XWZ= m\angle EWX+m\angle EYX+m\angle EYZ+m\angle EWZ=</math><math>360^\circ-m\angle CED-m\angle AEB=180^\circ</math>. |
<br/> | <br/> | ||
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<P align="right"><math>\mathbb{Q.E.D}</math></P> | <P align="right"><math>\mathbb{Q.E.D}</math></P> | ||
− | == | + | == See Also == |
− | {{USAMO box|year=1993|num-b= | + | {{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}} | ||
+ | |||
+ | [[Category:Olympiad Geometry Problems]] |
Latest revision as of 07:54, 19 July 2016
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.
See Also
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.