Difference between revisions of "2012 JBMO Problems/Problem 2"
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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 <math>k_1</math> and <math>k_2</math> that touches <math>k_1</math> and <math>k_2</math> at <math>M</math> and <math>N</math> respectively. If <math>t\perp AM</math> and <math>MN=2AM</math>, evaluate the angle <math>NMB</math>. | 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 <math>k_1</math> and <math>k_2</math> that touches <math>k_1</math> and <math>k_2</math> at <math>M</math> and <math>N</math> respectively. If <math>t\perp AM</math> and <math>MN=2AM</math>, evaluate the angle <math>NMB</math>. | ||
Latest revision as of 15:47, 11 January 2021
Problem
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 . Note that is a right angle since it inscribes a diameter. By AA similarity, . This gives that .
By Power of a Point on point with respect to circle , we have that . Using Power of a Point on point with respect to circle gives that . Therefore and . Since , . We now see that is a triangle. Since it is similar to , .
Solution by Someonenumber011 :)
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