Difference between revisions of "2018 IMO Problems/Problem 6"
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<i><b>Common case </b></i> | <i><b>Common case </b></i> | ||
− | Denote by <math>O</math> the intersection point of the | + | Denote by <math>O</math> the intersection point of the perpendicular bisector of <math>AC</math> and <math>BD.</math> Let <math>\omega</math> be a circle (red) with center <math>O</math> and radius <math>OA.</math> |
The points <math>B</math> and <math>D</math> are symmetric with respect to the circle <math>\omega</math> <i><b>(Claim 1).</b></i> | The points <math>B</math> and <math>D</math> are symmetric with respect to the circle <math>\omega</math> <i><b>(Claim 1).</b></i> | ||
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Circles <math>ACF</math> and <math>ACE</math> are symmetric with respect to the circle <math>\omega</math> <i><b>(Lemma).</b></i> | Circles <math>ACF</math> and <math>ACE</math> are symmetric with respect to the circle <math>\omega</math> <i><b>(Lemma).</b></i> | ||
+ | |||
+ | Denote by <math>X0</math> the point of intersection of the circles <math>BDF</math> and <math>ACE.</math> | ||
+ | Quadrangle <math>BX_0DF</math> is cyclic, hence, <math>\angle X_0BC = \angle X_0DA.</math> | ||
+ | Quadrangle <math>AX_0CE</math> is cyclic, hence, <math>\angle X_0AB = \angle X_0CD = \alpha.</math> | ||
+ | The required point <math>X = X_0</math> is constructed. |
Revision as of 07:40, 19 August 2022
A convex quadrilateral satisfies
Point
lies inside
so that
and
Prove that
Solution
Special case
We construct point and prove that
coincides with the point
Let and
Let and
be the intersection points of
and
and
and
respectively.
The points and
are symmetric with respect to the circle
(Claim 1).
The circle is orthogonal to the circle
(Claim 2).
Let be the point of intersection of the circles
and
Quadrilateral
is cyclic
Analogically, quadrangle is cyclic
.
This means that point coincides with the point
.
of
of
The sum (Claim 3)
Similarly,
Claim 1 Let and
be arbitrary points on a circle
be the middle perpendicular to the segment
Then the straight lines
and
intersect
at the points
and
symmetric with respect to
Claim 2 Let points and
be symmetric with respect to the circle
Then any circle
passing through these points is orthogonal to
Claim 3 The sum of the arcs between the points of intersection of two perpendicular circles is
In the figure they are a blue and red arcs
Common case
Denote by the intersection point of the perpendicular bisector of
and
Let
be a circle (red) with center
and radius
The points and
are symmetric with respect to the circle
(Claim 1).
The circles and
are orthogonal to the circle
(Claim 2).
Circles and
are symmetric with respect to the circle
(Lemma).
Denote by the point of intersection of the circles
and
Quadrangle
is cyclic, hence,
Quadrangle
is cyclic, hence,
The required point
is constructed.