Difference between revisions of "1993 USAMO Problems/Problem 2"
(Created page with '==Problem 2== Let <math>ABCD</math> be a convex quadrilateral such that diagonals <math>AC</math> and <math>BD</math> intersect at right angles, and let <math>E</math> be their …') |
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− | Let <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> be the foot of the altitute from point E of <math>\triangle AEB</math>, <math>\triangle BEC</math>, <math>\triangle CED</math>, <math>\triangle DEA</math>. | + | Let <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> be the foot of the altitute from point <math>E</math> of <math>\triangle AEB</math>, <math>\triangle BEC</math>, <math>\triangle CED</math>, <math>\triangle DEA</math>. |
Note that reflection of <math>E</math> over the 4 lines is <math>XYZW</math> with a scale of <math>2</math> with center <math>E</math>. Thus, if <math>XYZW</math> is cyclic, then the reflections are cyclic. | Note that reflection of <math>E</math> over the 4 lines is <math>XYZW</math> with a scale of <math>2</math> with center <math>E</math>. Thus, if <math>XYZW</math> is cyclic, then the reflections are cyclic. | ||
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<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=360^\circ-m\angle CED-m\angle AEB=180^\circ</math>. | ||
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Thus, <math>\angle XYZ</math> and <math>\angle XWZ</math> are supplementary and follows that, <math>XYZW</math> is cyclic. | Thus, <math>\angle XYZ</math> and <math>\angle XWZ</math> are supplementary and follows that, <math>XYZW</math> is cyclic. | ||
Revision as of 22:08, 22 April 2010
Contents
[hide]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 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |