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Difference between revisions of "2011 IMO Problems/Problem 6"

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The equation for line <math>AB</math> is <math>\frac{x-x_a}{y-y_a}=\frac{x-x_b}{y-y_b}</math>
 
The equation for line <math>AB</math> is <math>\frac{x-x_a}{y-y_a}=\frac{x-x_b}{y-y_b}</math>
  
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Through a little bash, (0,1) reflects to <math>(\frac{(-x_a+x_b+x_by_a)(y_b-_a)}{(x_a-x_b)^2+(y_a-y_b)^2}, \frac{(x_by_a-x_ay_b)(x_a-x_b)+(y_a-y_b)^2}{(x_a-x_b)^2+(y_a-y_b)^2}</math>. <math>(x_a-x_b)^2+(y_a-y_b)^2</math> simplifies to <math>2(1-x_ax_b-y_ay_b)</math>. The terms for the other 2 are symmetric.
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Through a little bash, (0,1) reflects to <math>\left(\frac{(-x_a+x_b+x_by_a)(y_b-_a)}{(x_a-x_b)^2+(y_a-y_b)^2}, \frac{(x_by_a-x_ay_b)(x_a-x_b)+(y_a-y_b)^2}{(x_a-x_b)^2+(y_a-y_b)^2}\right)</math>. <math>(x_a-x_b)^2+(y_a-y_b)^2</math> simplifies to <math>-(2x_ax_b+2y_ay_b)</math>. The terms for the other 2 are symmetric.
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The intersection point must reflect to itself, and the equation is <math>(\frac{x_by_a-x_ay_b-x_a+x_b}{y_b-y_a}, 1)</math>.
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The intersection point must reflect to itself, and the equation is <math>\left(\frac{x_by_a-x_ay_b-x_a+x_b}{y_b-y_a}, 1\right)</math>.
  
 
It is trivial to find the intersections of a,b and their perpendicular bisectors, so this is left to the reader as an exercise.
 
It is trivial to find the intersections of a,b and their perpendicular bisectors, so this is left to the reader as an exercise.
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==See Also==
 
==See Also==
 
*[[2011 IMO Problems]]
 
*[[2011 IMO Problems]]
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{{IMO box|year=2011|num-b=5|after=Last Problem}}

Latest revision as of 00:22, 19 November 2023

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.

Solution

Without loss of generality, let $\Gamma$ be the unit circle and let $\ell$ be the line $y=1$.

Denote the coordinates of $A,B,C$ by $(x_a, y_a)$ and similarly for B and C.

We get $x_a^2+y_a^2=1$, ...

The equation for line $AB$ is $\frac{x-x_a}{y-y_a}=\frac{x-x_b}{y-y_b}$

Through a little bash, (0,1) reflects to $\left(\frac{(-x_a+x_b+x_by_a)(y_b-_a)}{(x_a-x_b)^2+(y_a-y_b)^2}, \frac{(x_by_a-x_ay_b)(x_a-x_b)+(y_a-y_b)^2}{(x_a-x_b)^2+(y_a-y_b)^2}\right)$. $(x_a-x_b)^2+(y_a-y_b)^2$ simplifies to $-(2x_ax_b+2y_ay_b)$. The terms for the other 2 are symmetric. The intersection point must reflect to itself, and the equation is $\left(\frac{x_by_a-x_ay_b-x_a+x_b}{y_b-y_a}, 1\right)$.

It is trivial to find the intersections of a,b and their perpendicular bisectors, so this is left to the reader as an exercise.

Regardless, the circumcenter and an intersection of the circles are collinear with (0,0), so it is a tangency.

-Trex4days

See Also

2011 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
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