Difference between revisions of "1998 IMO Problems/Problem 1"
(Created page with "In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P , where the perpendicul...") |
(→See Also) |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular | + | ==Problem== |
| − | 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 | + | 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. |
| − | that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP | + | |
| − | have equal areas. | + | ==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
This problem needs a solution. If you have a solution for it, please help us out by adding it.
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 | ||
