Difference between revisions of "2017 IMO Problems/Problem 4"
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Quadrangle <math>AJST</math> is cyclic <math>\implies \angle RSJ = \angle TAJ \implies AT||RK.</math> | Quadrangle <math>AJST</math> is cyclic <math>\implies \angle RSJ = \angle TAJ \implies AT||RK.</math> | ||
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We construct circle <math>\omega</math> centered at <math>R</math> which maps <math>\Gamma</math> into <math>\Gamma.</math> | We construct circle <math>\omega</math> centered at <math>R</math> which maps <math>\Gamma</math> into <math>\Gamma.</math> |
Revision as of 01:55, 2 September 2022
Let and
be different points on a circle
such that
is not a diameter. Let
be the tangent line to
at
. Point
is such that
is the midpoint of the line segment
. Point
is chosen on the shorter arc
of
so that the circumcircle
of triangle
intersects
at two distinct points. Let
be the common point of
and
that is closer to
. Line
meets
again at
. Prove that the line
is tangent to
.
Solution
We construct inversion which maps into the circle
and
into
Than we prove that
is tangent to
Quadrangle is cyclic
Quadrangle is cyclic
We construct circle centered at
which maps
into
Let Inversion with respect to
swap
and
maps into
Let be the center of
Inversion with respect to maps
into
.
belong
circle
is the image of
. Let
be the center of
is the image of
at this inversion,
is tangent line to
at
so
is image K at this inversion
is parallelogram.
is the midpoint of
is the center of symmetry of
is symmetrical to
with respect to
is symmetrical to
with respect to
is symmetrycal
with respect to
lies on
and on
is tangent to
line
is tangent to
vladimir.shelomovskii@gmail.com, vvsss
Solution 2
We use the tangent-chord theorem: the angle formed between a chord and a tangent line to a circle is equal to the inscribed angle on the other side of the chord.
Quadrungle is cyclic
Quadrungle is cyclic
Let be symmetric to
with respect to
is parallelogram.
is cyclic.
Inscribed angle of is equal to angle between
and chord
is tangent to
by the inverse of tangent-chord theorem.
vladimir.shelomovskii@gmail.com, vvsss