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
,
, so
, and
.
, so
, and
.
$\angle PLK=\frac12(\arc{AD}+\arc{CF}=\frac12(\arc{CE}+\arc{AB}=\angle PKL$ (Error compiling LaTeX. Unknown error_msg), sois 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 |