Difference between revisions of "2017 IMO Problems/Problem 4"
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'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Solution 2== | ||
+ | [[File:2017 IMO 4a.png|500px|right]] | ||
+ | We use the equality of the inscribed angle and the angle between the chord and the tangent to prove that a line is a tangent. | ||
+ | |||
+ | Quadrungle <math>RJSK</math> is cyclic <math>\implies \angle RSJ = \angle RKJ.</math> | ||
+ | Quadrungle <math>AJST</math> is cyclic <math>\implies \angle RSJ = \angle TAJ \implies AT||RK.</math> | ||
+ | Let <math>B</math> be symmetric to <math>A</math> with respect to <math>S \implies ATBR</math> is parallelogram. | ||
+ | <math>\angle KST = \angle SAK + \angle SKR = \angle KRA</math> | ||
+ | <math>\angle RBT = \angle RAT \implies \angle KST + \angle KBT = 180^\circ \implies SKBT</math> is cyclic. | ||
+ | <math>\angle SBK = \angle STK = \angle SAT \implies </math> | ||
+ | Inscribed angle of <math>\Gamma \angle SAT</math> is equal to angle between KT and chord ST \implies <math>KT</math> is tangent to <math>\Gamma.</math> |
Revision as of 15:25, 26 August 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
Quadrungle is cyclic
Quadrungle
is cyclic
We construct circle centered at
which maps
into
Let Inversion with respect
swap
and
maps into
Let be the center of
Inversion with respect 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 equality of the inscribed angle and the angle between the chord and the tangent to prove that a line is a tangent.
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 KT and chord ST \implies
is tangent to