2016 AIME I Problems/Problem 15
Contents
Problem
Circles and intersect at points and . Line is tangent to and at and , respectively, with line closer to point than to . Circle passes through and intersecting again at and intersecting again at . The three points , , are collinear, , , and . Find .
Solution
Using the radical axis theorem, the lines are all concurrent at one point, call it . Now recall by Miquel's theorem in the fact that quadrilaterals and are cyclic implies is cyclic as well. Denote and .
Since point lies on the radical axis of , it has equal power with respect to both circles, thus Also, notice that The diagonals of quadrilateral bisect each other at , so we conclude that is a parallelogram. Let , so that .
Because is a parallelogram and quadrilaterals are cyclic, so we have the pair of similar triangles . Thus Now compute
Solution 1
Let . By the radical axis theorem are concurrent, say at . Moreover, by simple angle chasing. Let . Then Now, , and by power of a point, Solving, we get
Solution 2
By the Radical Axis Theorem concur at point .
Let and intersect at . Note that because and are cyclic, by Miquel's Theorem is cyclic as well. Thus and Thus and , so is a parallelogram. Hence and . But notice that and are similar by Similarity, so . But Hence
Solution 3
First, we note that as and have bases along the same line, . We can also find the ratio of their areas using the circumradius area formula. If is the radius of and if is the radius of , then Since we showed this to be , we see that .
We extend and to meet at point , and we extend and to meet at point as shown below. As is cyclic, we know that . But then as is tangent to at , we see that . Therefore, , and . A similar argument shows . These parallel lines show . Also, we showed that , so the ratio of similarity between and is , or rather We can now use the parallel lines to find more similar triangles. As , we know that Setting , we see that , hence , and the problem simplifies to finding . Setting , we also see that , hence . Also, as , we find that As , we see that , hence .
Applying Power of a Point to point with respect to , we find or . We wish to find .
Applying Stewart's Theorem to , we find We can cancel from both sides, finding . Therefore,
Solution 4
First of all, since quadrilaterals and are cyclic, we can let , and , due to the properties of cyclic quadrilaterals. In addition, let and . Thus, and . Then, since quadrilateral is cyclic as well, we have the following sums: Cancelling out in the second equation and isolating yields . Substituting back into the first equation, we obtain Since we can then imply that . Similarly, . So then , so since we know that bisects , we can solve for and with Stewart’s Theorem. Let and . Then Now, since and , . From there, let and . From angle chasing we can derive that and . From there, since , it is quite clear that , and can be found similarly. From there, since and , we have similarity between , , and . Therefore the length of is the geometric mean of the lengths of and (from ). However, yields the proportion ; hence, the length of is the geometric mean of the lengths of and . We can now simply use arithmetic to calculate .
-Solution by TheBoomBox77
Solution 5 (not too different)
Let . By Radical Axes, lies on . Note that is cyclic as is the Miquel point of in this configuration.
Claim. Proof. We angle chase. and
Let . Note andBy our claim, andFinally, ~Mathscienceclass
Solution 6 (No words)
vladimir.shelomovskii@gmail.com, vvsss
Solution 7 (Linearity of Power of a Point)
Extend and to meet at point . Let be the midpoint of segment . Then by radical axis on , and , lies on . By the bisector lemma, lies on . It is well-known that , , , and are concyclic. By Power of a point on with respect to and , so . Thus and bisect each other, so is a parallelogram. This implies that so by the inscribed angle theorem bisects .
Claim: .
Proof. Define the linear function . Since is parallel to the radical axis of and by our previous parallelism, . Note that while , so we conclude.
By Stewart's theorem on , , so .
~ Leo.Euler
Video Solution by MOP 2024
~r00tsOfUnity
Video Solution
~MathProblemSolvingSkills.com
Video Solution by The Power of Logic
See Also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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