# Difference between revisions of "2012 JBMO Problems/Problem 2"

(→Solution) |
(→Solution) |
||

Line 9: | Line 9: | ||

draw((0,0)--(2,2)); | draw((0,0)--(2,2)); | ||

draw((2,2)--(1,1)); | draw((2,2)--(1,1)); | ||

+ | draw((0,0)--(4,2)); | ||

draw(circle((0,1),1)); | draw(circle((0,1),1)); | ||

draw(circle((4,-3),5)); | draw(circle((4,-3),5)); | ||

Line 24: | Line 25: | ||

label("$O_1$",(0,1),NW); | label("$O_1$",(0,1),NW); | ||

label("$O_2$",(4,-3),NE); | label("$O_2$",(4,-3),NE); | ||

− | label("$k_1$",(-0.7,1. | + | label("$k_1$",(-0.7,1.63),NW); |

label("$k_2$",(7.6,0.46),NE); | label("$k_2$",(7.6,0.46),NE); | ||

label("$t$",(7.5,2),N); | label("$t$",(7.5,2),N); | ||

Line 30: | Line 31: | ||

</asy> | </asy> | ||

− | Let <math>O_1</math> and <math> | + | Let <math>O_1</math> and <math>O_2</math> be the centers of circles <math>k_1</math> and <math>k_2</math> respectively. Also let <math>P</math> be the intersection of <math>\overrightarrow{AB}</math> and line <math>t</math>. |

+ | |||

+ | Note that <math>\overline{O_1M}</math> is perpendicular to <math>\overline{MN}</math> since <math>M</math> is a tangent of <math>k_1</math>. In order for <math>\overline{AM}</math> to be perpendicular to <math>\overline{MN}</math>, <math>A</math> must be the point diametrically opposite <math>M</math>. (to be continued.) |

## Revision as of 22:32, 22 December 2020

## Section 2

Let the circles and intersect at two points and , and let be a common tangent of and that touches and at and respectively. If and , evaluate the angle .

## Solution

Let and be the centers of circles and respectively. Also let be the intersection of and line .

Note that is perpendicular to since is a tangent of . In order for to be perpendicular to , must be the point diametrically opposite . (to be continued.)