Difference between revisions of "2004 IMO Problems/Problem 5"
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In a convex quadrilateral <math>ABCD</math>, the diagonal <math>BD</math> bisects neither the angle <math>ABC</math> | In a convex quadrilateral <math>ABCD</math>, the diagonal <math>BD</math> bisects neither the angle <math>ABC</math> | ||
nor the angle <math>CDA</math>. The point <math>P</math> lies inside <math>ABCD</math> and satisfies <cmath>\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.</cmath> | nor the angle <math>CDA</math>. The point <math>P</math> lies inside <math>ABCD</math> and satisfies <cmath>\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.</cmath> | ||
Prove that <math>ABCD</math> is a cyclic quadrilateral if and only if <math>AP = CP.</math> | Prove that <math>ABCD</math> is a cyclic quadrilateral if and only if <math>AP = CP.</math> | ||
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| + | ==Solution== | ||
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| + | ==See Also== | ||
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| + | {{IMO box|year=2004|num-b=4|num-a=6}} | ||
Revision as of 23:54, 18 November 2023
Problem
In a convex quadrilateral 
, the diagonal 
bisects neither the angle 
nor the angle 
. The point 
lies inside and satisfies ![\[\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.\]](http://latex.artofproblemsolving.com/9/7/7/977f696cf8654292a2d57df388ce9a1baff926af.png)
Prove that is a cyclic quadrilateral if and only if 
Solution
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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 | ||
