2019 AIME II Problems/Problem 11
Triangle has side lengths and Circle passes through and is tangent to line at Circle passes through and is tangent to line at Let be the intersection of circles and not equal to Then where and are relatively prime positive integers. Find
-Diagram by Brendanb4321
Note that from the tangency condition that the supplement of with respects to lines and are equal to and , respectively, so from tangent-chord, Also note that , so . Using similarity ratios, we can easily find However, since and , we can use similarity ratios to get
- Now we use Law of Cosines on : From reverse Law of Cosines,
Giving us so our answer is .
Let be the center of . Then . Thus,
-franchester; by firebolt360
Solution 2 (Inversion)
Consider an inversion with center and radius . Then, we have , or . Similarly, . Notice that is a parallelogram, since and are tangent to and , respectively. Thus, . Now, we get that so by Law of Cosines on we have Then, our answer is . -brianzjk
Solution 3 (Death By Trig Bash)
Let the centers of the circles be and where the has the side length contained in the circle. Now let This implies by the angle by by tangent. Then we also know that Now we first find We use law of cosines on to obtain Then applying law of sines on we obtain Using similar logic we obtain
Now we know that Thus using law of cosines on yields While this does look daunting we can write the above expression as Then factoring yields The area Now is twice the length of the altitude of so we let the altitude be and we have Thus our desired length is
Solution 4 (Video)
Video Link: https://www.youtube.com/watch?v=nJydO5CLuuI
Solution 5 (Olympiad Geometry)
By the definition of , it is the spiral center mapping , which means that it is the midpoint of the -symmedian chord. In particular, if is the midpoint of and is the reflection of across , we have . By Stewart's Theorem, it then follows that
Solution 6 (Inversion simplified)
The median of is (via Stewart's Theorem).
Consider an inversion with center and radius (inversion with respect the red circle). Let and be inverse points for and respectively.
Image of line is line lies on this line.
Image of is line (circle passes through K, C and is tangent to the line at point Diagram shows circle and its image using same color).
Similarly, is the image of the circle ).
Therefore is a parallelogram, is median of and Then, we have . with coefficient
So median email@example.com, vvsss
Solution 7 (Heavy Bash)
We start by assigning coordinates to point , labeling it and point at , and letting point be above the -axis. Through an application of the Pythagorean Theorem and dropping an altitude to side , it is easy to see that has coordinates .
Let be the center of circle and be the center of circle . Since circle contains both points and , must lie on the perpendicular bisector of line , and similarly must lie on the perpendicular bisector of line . Through some calculations, we find that the perpendicular bisector of has equation , and the perpendicular bisector of has equation .
Since circle is tangent to line at , its radius must be perpendicular to at . Therefore, the radius has equation . Substituting the -coordinate of into this, we find the y-coordinate of .
Similarly, since circle is tangent to line at , its radius must be perpendicular to at . Therefore, the radius has equation and combining with the previous result for we get that the coordinates of are .
We now find the slope of , the line joining the centers of circles and , which turns out to be . Since the -intercept of that line is at , the equation is . Since circles and intersect at points and , line is the radical axis of those circles, and since the radical axis is always perpendicular to the line joining the centers of the circles, has slope . Since point is , this line has a -intercept of , so it has equation = .
We set in order to find the intersection of the radical axis and . Through some moderate bashing, we find that the intersection point is . We know that either intersection point of two circles is the same distance from the intersection of radical axis and line joining the centers of those circles, so reflecting over yields and = = (This is the most tedious part of the bash) . Therefore the answer is
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