Difference between revisions of "2004 IMO Problems/Problem 5"
Szhangmath (talk | contribs) (→Solution) |
Szhangmath (talk | contribs) (→Solution) |
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Let <math>K</math> be the intersection of <math>AC</math> and <math>BE</math>, let <math>L</math> be the intersection of <math>AC</math> and <math>DF</math>, | Let <math>K</math> be the intersection of <math>AC</math> and <math>BE</math>, let <math>L</math> be the intersection of <math>AC</math> and <math>DF</math>, | ||
+ | [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("<math>A</math>", (-6.8,-2), SW); | ||
+ | label("<math>B</math>", (-3.7,-6), SW); | ||
+ | label("<math>F</math>", (3.7,-6), SE); | ||
+ | label("<math>C</math>", (6.8,-2), E); | ||
+ | label("<math>E</math>", (5,5), E); | ||
+ | label("<math>D</math>", (-5,5), W); | ||
+ | label("<math>P</math>", (0,-1.3), N); | ||
+ | label("<math>K</math>", (-1,-1.6), E); | ||
+ | label("<math>L</math>", (0.7,-1.6) ); | ||
+ | ~ | ||
+ | [/asy] | ||
<math>\angle PBC=\angle DBA</math>, so <math>AD=CE</math>, and <math>DE//AC</math>. | <math>\angle PBC=\angle DBA</math>, so <math>AD=CE</math>, and <math>DE//AC</math>. | ||
<math>\angle PDC=\angle BDA</math>, so <math>AB=CF</math>, and <math>AC//BF</math>. | <math>\angle PDC=\angle BDA</math>, so <math>AB=CF</math>, and <math>AC//BF</math>. |
Revision as of 16:11, 8 February 2024
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
This problem needs a solution. If you have a solution for it, please help us out by adding it.
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 |