Difference between revisions of "1998 IMO Problems/Problem 1"
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==Problem== | ==Problem== | ||
| − | In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular | + | In the convex quadrilateral <math>ABCD</math>, the diagonals <math>AC</math> and <math>BD</math> are perpendicular and the opposite sides <math>AB</math> and <math>DC</math> are not parallel. Suppose that the point <math>P</math>, where the perpendicular bisectors of <math>AB</math> and <math>DC</math> meet, is inside <math>ABCD</math>. Prove that <math>ABCD</math> is a cyclic quadrilateral if and only if the triangles <math>ABP</math> and <math>CDP</math> have equal areas. |
| − | and the opposite sides AB and DC are not parallel. Suppose that the point P , | ||
| − | where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove | ||
| − | that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP | ||
| − | have equal areas. | ||
==Solution== | ==Solution== | ||
{{solution}} | {{solution}} | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{IMO box|year=1998|before=First Question|num-a=2}} | ||
Latest revision as of 23:53, 18 November 2023
Problem
In the convex quadrilateral 
, the diagonals 
and 
are perpendicular and the opposite sides 
and 
are not parallel. Suppose that the point 
, where the perpendicular bisectors of and meet, is inside . Prove that is a cyclic quadrilateral if and only if the triangles 
and 
have equal areas.
Solution
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See Also
| 1998 IMO (Problems) • Resources | ||
| Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
