Difference between revisions of "2019 IMO Problems/Problem 2"
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Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic. | Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic. | ||
+ | |||
+ | ==Solution== | ||
+ | [[File:2019 IMO 2.png|500px|right]] | ||
+ | The essence of the proof is to build a circle through the points <math>P, Q,</math> and two additional points <math>A_0</math> and <math>B_0,</math> then we prove that the points <math>P_1</math> and <math>Q_1</math> lie on the same circle. | ||
+ | |||
+ | Let the circumcircle of <math>\triangle ABC</math> be <math>\Omega</math>. Let <math>A_0</math> and <math>B_0</math> be the points of intersection of <math>AP</math> and <math>BQ</math> with <math>\Omega</math>. Let <math>\angle BAP = \delta.</math> | ||
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+ | <cmath>PQ||AB \implies \angle QPA_0 = \delta.</cmath> | ||
+ | |||
+ | <math>\angle BAP = \angle BB_0A_0 = \delta</math> since they intersept the arc <math>BA_0</math> of the circle <math>\Omega</math>. | ||
+ | |||
+ | <math>\angle QPA_0 = \angle QB_0A_0 \implies QPB_0A_0</math> is cyclic (in circle <math>\omega.</math>) |
Revision as of 11:53, 13 August 2022
In triangle , point
lies on side
and point
lies on side
. Let
and
be points on segments
and
, respectively, such that
is parallel to
. Let
be a point on line
, such that
lies strictly between
and
, and
. Similarly, let
be the point on line
, such that
lies strictly between
and
, and
.
Prove that points , and
are concyclic.
Solution
The essence of the proof is to build a circle through the points and two additional points
and
then we prove that the points
and
lie on the same circle.
Let the circumcircle of be
. Let
and
be the points of intersection of
and
with
. Let
since they intersept the arc
of the circle
.
is cyclic (in circle
)