Difference between revisions of "1993 USAMO Problems/Problem 2"
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== Solution 2 == | == Solution 2 == | ||
− | Suppose the reflection of E over AB is W, and similarly define X, Y, and Z. | + | Suppose the reflection of E over AB is W, and similarly define X, Y, and Z. |
− | <math>\bigtriangleup BEA \cong \bigtriangleup BWA</math> by reflection gives <math>BE = BW</math> | + | <math>\bigtriangleup BEA \cong \bigtriangleup BWA</math> by reflection gives <math>BE = BW</math> |
− | <math>\bigtriangleup BEC \cong \bigtriangleup BXC</math> by reflection gives <math>BE = BX</math> | + | <math>\bigtriangleup BEC \cong \bigtriangleup BXC</math> by reflection gives <math>BE = BX</math> |
− | These two tell us that E, W, and X belong to a circle with center B. | + | These two tell us that E, W, and X belong to a circle with center B. |
− | Similarly, we can get that: | + | Similarly, we can get that: |
− | E, Z, and W belong to a circle with center A, | + | E, Z, and W belong to a circle with center A, |
− | E, X, and Y belong to a circle with center C, | + | E, X, and Y belong to a circle with center C, |
− | E, Y, and Z belong to a circle with center D. | + | E, Y, and Z belong to a circle with center D. |
− | + | ||
− | To prove that W, X, Y, Z are concyclic, we want to prove <math>\angle XWZ + \angle XYZ = 180^o</math> | + | To prove that W, X, Y, Z are concyclic, we want to prove <math>\angle XWZ + \angle XYZ = 180^o</math> |
− | <math>\angle XWZ + \angle XYZ = \angle XWE + \angle EWZ + \angle XYE + \angle EYZ</math> | + | <math>\angle XWZ + \angle XYZ = \angle XWE + \angle EWZ + \angle XYE + \angle EYZ</math> |
− | <math> = \frac{1}{2} \angle XBE + \frac{1}{2} \angle EAZ + \frac{1}{2} \angle XCE + \frac{1}{2} \angle EDZ</math> | + | <math> = \frac{1}{2} \angle XBE + \frac{1}{2} \angle EAZ + \frac{1}{2} \angle XCE + \frac{1}{2} \angle EDZ</math> |
− | <math> = \frac{1}{2} (\angle XBE + \angle XCE) + \frac{1}{2} (\angle EAZ + \angle EDZ)</math> | + | <math> = \frac{1}{2} (\angle XBE + \angle XCE) + \frac{1}{2} (\angle EAZ + \angle EDZ)</math> |
− | + | ||
− | <math>\angle AED = 90^o</math> and <math>\angle AED = \angle AZD</math> tells us that <math>\angle EAZ + \angle EDZ = 180^o</math> | + | <math>\angle AED = 90^o</math> and <math>\angle AED = \angle AZD</math> tells us that <math>\angle EAZ + \angle EDZ = 180^o</math> |
− | Similarly, <math>\angle XBE + \angle XCE = 180^o</math> | + | Similarly, <math>\angle XBE + \angle XCE = 180^o</math> |
− | Thus, <math>\angle XWZ + \angle XYZ = \frac{1}{2} \cdot 180^o + \frac{1}{2} \cdot 180^o = 180^o</math>, and we are done. | + | Thus, <math>\angle XWZ + \angle XYZ = \frac{1}{2} \cdot 180^o + \frac{1}{2} \cdot 180^o = 180^o</math>, and we are done. |
-- Lucas.xue (someone pls help with a diagram) | -- Lucas.xue (someone pls help with a diagram) | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | E lies on the isoptic cubic of ABCD, so it has an isogonal conjugate in ABCD. | ||
Latest revision as of 04:42, 1 October 2024
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 1
Diagram
Work
Let , , , be the foot of the altitude from point of , , , .
Note that reflection of over all the points of is similar to 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.
Solution 2
Suppose the reflection of E over AB is W, and similarly define X, Y, and Z. by reflection gives by reflection gives These two tell us that E, W, and X belong to a circle with center B. Similarly, we can get that: E, Z, and W belong to a circle with center A, E, X, and Y belong to a circle with center C, E, Y, and Z belong to a circle with center D.
To prove that W, X, Y, Z are concyclic, we want to prove
and tells us that Similarly, Thus, , and we are done. -- Lucas.xue (someone pls help with a diagram)
Solution 3
E lies on the isoptic cubic of ABCD, so it has an isogonal conjugate in ABCD.
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.