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Difference between revisions of "Sharygin Olympiads, the best"

(→‎2024, Problem 23)
(→‎2024, Problem 23)
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==2024, Problem 23==
 
==2024, Problem 23==
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[[File:2023 22 2.png|350px|right]]
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[[File:2023 23 1.png|350px|right]]
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A point <math>P</math> moves along a circle <math>\Omega.</math> Let <math>A</math> and <math>B</math> be fixed points of <math>\Omega,</math> and <math>C</math> be an arbitrary point inside <math>\Omega.</math> The common external tangents to the circumcircles of triangles <math>\triangle APC</math> and <math>\triangle BCP</math> meet at point <math>Q.</math> Prove that all points <math>Q</math> lie on two fixed lines.
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A point <math>P</math> moves along a circle <math>\Omega.</math> Let <math>A</math> and <math>B</math> be fixed points of <math>\Omega,</math> and <math>C</math> be an arbitrary point inside <math>\Omega.</math>
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The common external tangents to the circumcircles of triangles <math>\triangle APC</math> and <math>\triangle BCP</math> meet at point <math>Q.</math>
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Prove that all points <math>Q</math> lie on two fixed lines.
  
 
<i><b>Solution</b></i>
 
<i><b>Solution</b></i>
  
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Denote <math>A' = AC \cup \Omega, B' = BC \cup \Omega, \omega = \odot APC, \omega' = \odot BPC,</math>
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Denote <math>A' = AC \cap \Omega, B' = BC \cap \Omega, \omega = \odot APC, \omega' = \odot BPC.</math>
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<math>\theta = \odot ACB', \theta' = \odot BCA', O</math> is the circumcenter of <math>\triangle APC, O'</math> is the circumcenter of <math>\triangle BPC.</math>
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<math>\theta = \odot ACB', \theta' = \odot BCA'.</math>
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<math>O</math> is the circumcenter of <math>\triangle APC, O'</math> is the circumcenter of <math>\triangle BPC.</math>
 
   
 
   
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Let <math>K</math> and <math>L</math> be the midpoints of the arcs <math>CB'</math> of <math>\theta.</math>
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Let <math>K</math> and <math>L</math> be the midpoints of the arcs <math>\overset{\Large\frown}{CB'}</math> of <math>\theta.</math>
  
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Let <math>K'</math> and <math>L'</math> be the midpoints of the arcs <math>CA'</math> of <math>\theta'.</math> These points not depends from position of point <math>P.</math>
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Let <math>K'</math> and <math>L'</math> be the midpoints of the arcs <math>\overset{\Large\frown}{CA'}</math> of <math>\theta'.</math>
  
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Suppose, <math>P \in \overset{\Large\frown} {B'ABA'}.</math>
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These points not depends from position of point <math>P.</math>
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Suppose, <math>P \in \overset{\Large\frown} {B'ABA'} (</math> see diagram).
 
<cmath>\angle A'BC = 2 \alpha = \angle B'AC \implies \angle D'BC = \angle DAC = \alpha \implies \angle DOC = \angle D'O'C = 2 \alpha.</cmath>
 
<cmath>\angle A'BC = 2 \alpha = \angle B'AC \implies \angle D'BC = \angle DAC = \alpha \implies \angle DOC = \angle D'O'C = 2 \alpha.</cmath>
 
<cmath>O'D' = O'C, OC = OD \implies \triangle OCD \sim \triangle O'D'C \implies OC||O'D'.</cmath>
 
<cmath>O'D' = O'C, OC = OD \implies \triangle OCD \sim \triangle O'D'C \implies OC||O'D'.</cmath>
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Let <math>F= CD \cup OO' \implies \frac {FO}{FO'} = \frac {OC}{O'D'} \implies Q = F.</math>
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<cmath>\angle LCB' = \alpha = \angle B'BL' \implies LC || L'B.</cmath>
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Similarly, <math>AL || CL' \implies \triangle DLC \sim \triangle CL'D' \implies \frac {LC}{L'D'} = \frac {DC}{CD'} = \frac {OC}{O'D'}.</math>
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Let <math>F' = LL' \cap DD' \implies \frac {F'C}{F'D'} = \frac {LC}{L'D'} = \frac {OC}{O'D'}=  \frac {FC}{FD'}  \implies F' = F.</math>
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Therefore <math>Q \in LL'.</math> Similarly, if <math>P \in \overset{\Large\frown} {B'A'}</math> then <math>Q \in KK'.</math>
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'''vladimir.shelomovskii@gmail.com, vvsss'''

Revision as of 01:06, 24 March 2024

Igor Fedorovich Sharygin (13/02/1937 - 12/03/2004, Moscow) - Soviet and Russian mathematician and teacher, specialist in elementary geometry, popularizer of science. He wrote many textbooks on geometry and created a number of beautiful problems. He headed the mathematics section of the Russian Soros Olympiads. After his death, Russia annually hosts the Geometry Olympiad for high school students. It consists of two rounds – correspondence and final. The correspondence round lasts 3 months.

The best problems of these Olympiads will be published. The numbering contains the year of the Olympiad and the serial number of the problem. Solutions are often different from the original ones.

2024, Problem 23

2023 23 1.png

A point $P$ moves along a circle $\Omega.$ Let $A$ and $B$ be fixed points of $\Omega,$ and $C$ be an arbitrary point inside $\Omega.$

The common external tangents to the circumcircles of triangles $\triangle APC$ and $\triangle BCP$ meet at point $Q.$

Prove that all points $Q$ lie on two fixed lines.

Solution

Denote $A' = AC \cap \Omega, B' = BC \cap \Omega, \omega = \odot APC, \omega' = \odot BPC.$ $\theta = \odot ACB', \theta' = \odot BCA'.$

$O$ is the circumcenter of $\triangle APC, O'$ is the circumcenter of $\triangle BPC.$

Let $K$ and $L$ be the midpoints of the arcs $\overset{\Large\frown}{CB'}$ of $\theta.$

Let $K'$ and $L'$ be the midpoints of the arcs $\overset{\Large\frown}{CA'}$ of $\theta'.$

These points not depends from position of point $P.$

Suppose, $P \in \overset{\Large\frown} {B'ABA'} ($ see diagram). \[\angle A'BC = 2 \alpha = \angle B'AC \implies \angle D'BC = \angle DAC = \alpha \implies \angle DOC = \angle D'O'C = 2 \alpha.\] \[O'D' = O'C, OC = OD \implies \triangle OCD \sim \triangle O'D'C \implies OC||O'D'.\] Let $F= CD \cup OO' \implies \frac {FO}{FO'} = \frac {OC}{O'D'} \implies Q = F.$ \[\angle LCB' = \alpha = \angle B'BL' \implies LC || L'B.\] Similarly, $AL || CL' \implies \triangle DLC \sim \triangle CL'D' \implies \frac {LC}{L'D'} = \frac {DC}{CD'} = \frac {OC}{O'D'}.$

Let $F' = LL' \cap DD' \implies \frac {F'C}{F'D'} = \frac {LC}{L'D'} = \frac {OC}{O'D'}= \frac {FC}{FD'} \implies F' = F.$

Therefore $Q \in LL'.$ Similarly, if $P \in \overset{\Large\frown} {B'A'}$ then $Q \in KK'.$

vladimir.shelomovskii@gmail.com, vvsss

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