2004 IMO Problems/Problem 5
Problem
In a convex quadrilateral , the diagonal bisects neither the angle nor the angle . The point lies inside and satisfies
Prove that is a cyclic quadrilateral if and only if
Solution
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Let be the intersection of and , let be the intersection of and , [asy]
size(10cm); draw(circle((0,0),7.07)); draw((-3.7,-6)-- (3.7,-6)); draw((-6.8,-2)-- (6.8,-2)); draw((-5,5)-- (5,5)); draw((-5,5)-- (-3.7,-6)); draw((-5,5)-- (3.7,-6)); draw((-5,5)-- (-6.8,-2)); draw((-5,5)-- (6.8,-2));
draw((5,5)-- (-3.7,-6)); draw((5,5)-- (3.7,-6)); draw((5,5)-- (-6.8,-2)); draw((5,5)-- (6.8,-2)); draw((-3.7,-6)-- (-6.8,-2)); draw((-3.7,-6)-- (6.8,-2)); draw((3.7,-6)-- (-6.8,-2)); draw((3.7,-6)-- (6.8,-2)); label("", (-6.8,-2), SW); label("", (-3.7,-6), SW); label("", (3.7,-6), SE); label("", (6.8,-2), E); label("", (5,5), E); label("", (-5,5), W); label("", (0,-1.3), N); label("", (-1,-1.6), E); label("", (0.7,-1.6) ); ~ [/asy] , so , and . , so , and .
, so is an isosceles triangle. Since , so and are isosceles triangles. So is on the angle bisector oof , since is an isosceles trapezoid, so is also on the perpendicular bisector of . So .
~szhangmath
See Also
2004 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |