2012 JBMO Problems/Problem 2: Difference between revisions
Created page with "== Section 2== Let the circles <math>k_1</math> and <math>k_2</math> intersect at two points <math>A</math> and <math>B</math>, and let <math>t</math> be a common tangent of..." |
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== Solution == | == Solution == | ||
<asy> | |||
size(15cm,0); | |||
draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); | |||
draw((-1,2)--(9,2)); | |||
draw((0,0)--(2,2)); | |||
draw((2,2)--(1,1)); | |||
draw(circle((0,1),1)); | |||
draw(circle((4,-3),5)); | |||
dot((0,0)); | |||
dot((0,2)); | |||
dot((2,2)); | |||
dot((4,2)); | |||
dot((4,-3)); | |||
dot((1,1)); | |||
dot((0,1)); | |||
label("A",(0,0),NW); | |||
label("B",(1,1),NW); | |||
label("M",(0,2),N); | |||
label("N",(4,2),N); | |||
label("$O_1$",(0,1),NW); | |||
label("$O_2$",(4,-3),NE); | |||
label("$k_1$",(-0.7,1.7),NW); | |||
label("$k_2$",(7.6,0.46),NE); | |||
label("$t$",(7.5,2),N); | |||
label("P",(2,2),N); | |||
</asy> | |||
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. | |||
Revision as of 21:16, 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
![[asy] size(15cm,0); draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); draw((-1,2)--(9,2)); draw((0,0)--(2,2)); draw((2,2)--(1,1)); draw(circle((0,1),1)); draw(circle((4,-3),5)); dot((0,0)); dot((0,2)); dot((2,2)); dot((4,2)); dot((4,-3)); dot((1,1)); dot((0,1)); label("A",(0,0),NW); label("B",(1,1),NW); label("M",(0,2),N); label("N",(4,2),N); label("$O_1$",(0,1),NW); label("$O_2$",(4,-3),NE); label("$k_1$",(-0.7,1.7),NW); label("$k_2$",(7.6,0.46),NE); label("$t$",(7.5,2),N); label("P",(2,2),N); [/asy]](http://latex.artofproblemsolving.com/4/f/5/4f53e21aa632b2964a3cbc61b44f83406f259b9f.png)
Let 
and 
be the centers of circles and respectively.
