Difference between revisions of "2004 IMO Problems/Problem 5"
Szhangmath (talk | contribs) (→Solution) |
Szhangmath (talk | contribs) (→Solution) |
||
Line 43: | Line 43: | ||
<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>. | ||
<math>\angle PLK=\frac12(\overarc{AD}+\overarc{CF})=\frac12(\overarc{CE}+\overarc{AB})=\angle PKL</math>, so <math>\triangle PKL</math> is an isosceles triangle. | <math>\angle PLK=\frac12(\overarc{AD}+\overarc{CF})=\frac12(\overarc{CE}+\overarc{AB})=\angle PKL</math>, so <math>\triangle PKL</math> is an isosceles triangle. | ||
− | Since <math>AC//BF</math>, so <math>\triangle PBF</math> and <math>\triangle PDE</math> are isosceles triangles. So <math>P</math> is on the | + | Since <math>AC//BF</math>, so <math>\triangle PBF</math> and <math>\triangle PDE</math> are isosceles triangles. So <math>P</math> is on the perpendicular bisector of <math>BF</math>, since <math>ABFC</math> is |
an isosceles trapezoid, so <math>P</math> is also on the perpendicular bisector of <math>AC</math>. So <math>PA=PC</math>. | an isosceles trapezoid, so <math>P</math> is also on the perpendicular bisector of <math>AC</math>. So <math>PA=PC</math>. | ||
Revision as of 16:48, 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
,
, so
, and
.
, so
, and
.
, so
is an isosceles triangle. Since
, so
and
are isosceles triangles. So
is on the perpendicular bisector of
, 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 |