1993 USAMO Problems/Problem 2
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)
See Also
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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