Difference between revisions of "2009 OIM Problems/Problem 3"
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− | Let <math>C_1</math> and <math>C_2</math> be two circles with centers <math>O_1</math> and <math>O_2</math> with the same radius, which intersect at <math>A</math> and <math>B</math>. Let <math>P</math> be a point on the arc <math>AB</math> of <math>C_2</math> that is inside <math>C_1</math>. Line <math>AP</math> intersects <math>C_1</math> at <math>C</math>, line <math>CB</math> intersects <math>C_2</math> at <math>D</math> and the bisector of <math>\angle CAD</math> intersects <math>C_1</math> at <math>E</math> and <math>C_2</math> at <math>L</math>. Let <math>F</math> be the point symmetrical to <math>D</math> with respect to the midpoint of <math>PE</math>. Show that there exists a point <math>X</math> satisfying <math>\angle | + | Let <math>C_1</math> and <math>C_2</math> be two circles with centers <math>O_1</math> and <math>O_2</math> with the same radius, which intersect at <math>A</math> and <math>B</math>. Let <math>P</math> be a point on the arc <math>AB</math> of <math>C_2</math> that is inside <math>C_1</math>. Line <math>AP</math> intersects <math>C_1</math> at <math>C</math>, line <math>CB</math> intersects <math>C_2</math> at <math>D</math> and the bisector of <math>\angle CAD</math> intersects <math>C_1</math> at <math>E</math> and <math>C_2</math> at <math>L</math>. Let <math>F</math> be the point symmetrical to <math>D</math> with respect to the midpoint of <math>PE</math>. Show that there exists a point <math>X</math> satisfying <math>\angle XFL = \angle XCD = 30^{^circ}</math> and <math>CX = O_1O_2</math>. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com |
Revision as of 16:21, 14 December 2023
Problem
Let and
be two circles with centers
and
with the same radius, which intersect at
and
. Let
be a point on the arc
of
that is inside
. Line
intersects
at
, line
intersects
at
and the bisector of
intersects
at
and
at
. Let
be the point symmetrical to
with respect to the midpoint of
. Show that there exists a point
satisfying
and
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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