Difference between revisions of "2016 AIME I Problems/Problem 15"
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==Solution== | ==Solution== | ||
+ | Using the radical axis theorem, the lines <math>\overline{AD}, \overline{BC}, \overline{XY}</math> are all concurrent at one point, call it <math>F</math>. Now recall by Miquel's theorem in <math>\triangle FDC</math> the fact that quadrilaterals <math>DAXY</math> and <math>CBXY</math> are cyclic implies <math>FAXB</math> is cyclic as well. Denote <math>\omega_{3}\equiv(FAXB)</math> and <math>Z\equiv\ell\cap\overline{FXY}</math>. | ||
− | ===Solution 1=== | + | Since point <math>Z</math> lies on the radical axis of <math>\omega_{1},\omega_{2}</math>, it has equal power with respect to both circles, thus <cmath>AZ^{2}=\text{Pow}_{\omega_{1}}(Z)=ZX\cdot ZY=\text{Pow}_{\omega_{2}}(Z)=ZB^{2}\implies AZ=ZB.</cmath> Also, notice that <cmath>AZ\cdot ZB=\text{Pow}_{\omega_{3}}(Z)=ZX\cdot ZF\implies ZY=ZF.</cmath> The diagonals of quadrilateral <math>FAYB</math> bisect each other at <math>Z</math>, so we conclude that <math>FAYB</math> is a parallelogram. Let <math>u:=ZX</math>, so that <math>ZY=ZF=u+47</math>. |
+ | |||
+ | Because <math>FAYB</math> is a parallelogram and quadrilaterals <math>DAXY, CBXY</math> are cyclic, <cmath>\angle DFX=\angle AFX=\angle BYX=\angle BCX=\angle FCX~~\text{and}~~\angle XDF=\angle XDA=\angle XYA=\angle XFB=\angle XFC</cmath> so we have the pair of similar triangles <math>\triangle DFX~\sim~\triangle FCX</math>. Thus <cmath>\dfrac{37}{2u+47}=\dfrac{2u+47}{67}\implies 2u+47=\sqrt{37\cdot 67}\implies u=\dfrac{1}{2}\left(\sqrt{37\cdot 67}-47\right).</cmath> Now compute <cmath>AB^{2}=4AZ^{2}=4\cdot ZX\cdot ZY=4u(u+47)=37\cdot 67-47^{2}=\textbf{270}.</cmath> | ||
+ | |||
+ | [[File:AIME 2016-I15 Geogebra Diagram.png|840px]] | ||
+ | |||
+ | ==Solution 1== | ||
+ | |||
+ | Let <math>Z = XY \cap AB</math>. By the radical axis theorem <math>AD, XY, BC</math> are concurrent, say at <math>P</math>. Moreover, <math>\triangle DXP \sim \triangle PXC</math> by simple angle chasing. Let <math>y = PX, x = XZ</math>. Then <cmath>\frac{y}{37} = \frac{67}{y} \qquad \implies \qquad y^2 = 37 \cdot 67.</cmath> Now, <math>AZ^2 = \tfrac 14 AB^2</math>, and by power of a point, <cmath>\begin{align*} | ||
+ | x(y-x) &= \tfrac 14 AB^2, \quad \text{and} \\ | ||
+ | x(47+x) &= \tfrac 14 AB^2 | ||
+ | \end{align*}</cmath> Solving, we get <cmath>\tfrac 14 AB^2 = \tfrac 12 (y-47)\cdot \tfrac 12 (y+47) \qquad \implies</cmath> | ||
+ | <cmath> \qquad AB ^ 2 = 37\cdot 67 - 47^2 = \boxed{270}</cmath> | ||
+ | |||
+ | ==Solution 2== | ||
By the Radical Axis Theorem <math>AD, XY, BC</math> concur at point <math>E</math>. | By the Radical Axis Theorem <math>AD, XY, BC</math> concur at point <math>E</math>. | ||
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<cmath>XE^2 - XY^2 = (XE + XY)(XE - XY) = EY \cdot 2XS = 2SY \cdot 2SX = 4SA^2 = AB^2.</cmath>Hence <math>AB^2 = 37 \cdot 67 - 47^2 = \boxed{270}.</math> | <cmath>XE^2 - XY^2 = (XE + XY)(XE - XY) = EY \cdot 2XS = 2SY \cdot 2SX = 4SA^2 = AB^2.</cmath>Hence <math>AB^2 = 37 \cdot 67 - 47^2 = \boxed{270}.</math> | ||
− | + | ==Solution 3== | |
First, we note that as <math>\triangle XDY</math> and <math>\triangle XYC</math> have bases along the same line, <math>\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{DY}{YC}</math>. We can also find the ratio of their areas using the circumradius area formula. If <math>R_1</math> is the radius of <math>\omega_1</math> and if <math>R_2</math> is the radius of <math>\omega_2</math>, then | First, we note that as <math>\triangle XDY</math> and <math>\triangle XYC</math> have bases along the same line, <math>\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{DY}{YC}</math>. We can also find the ratio of their areas using the circumradius area formula. If <math>R_1</math> is the radius of <math>\omega_1</math> and if <math>R_2</math> is the radius of <math>\omega_2</math>, then | ||
<cmath>\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{(37\cdot 47\cdot DY)/(4R_1)}{(47\cdot 67\cdot YC)/(4R_2)}=\frac{37\cdot DY\cdot R_2}{67\cdot YC\cdot R_1}.</cmath> | <cmath>\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{(37\cdot 47\cdot DY)/(4R_1)}{(47\cdot 67\cdot YC)/(4R_2)}=\frac{37\cdot DY\cdot R_2}{67\cdot YC\cdot R_1}.</cmath> | ||
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<cmath>AB^2=30^2\cdot 37\cdot 67y^2=37\cdot 67-47^2=\boxed{270}.</cmath> | <cmath>AB^2=30^2\cdot 37\cdot 67y^2=37\cdot 67-47^2=\boxed{270}.</cmath> | ||
− | ==Solution | + | ==Solution 4== |
<asy> | <asy> | ||
size(9cm); | size(9cm); | ||
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<cmath>\theta + x +\angle XCY + y = 180^{\circ}</cmath> | <cmath>\theta + x +\angle XCY + y = 180^{\circ}</cmath> | ||
<cmath>180^{\circ} - \theta + y + \angle XDY + x = 180^{\circ}</cmath> | <cmath>180^{\circ} - \theta + y + \angle XDY + x = 180^{\circ}</cmath> | ||
− | Cancelling out <math>180^{\circ}</math> in the second equation isolating <math>\theta</math> yields <math>\theta = y + \angle XDY + x</math>. Substituting <math>\theta</math> back into the first equation, we obtain | + | Cancelling out <math>180^{\circ}</math> in the second equation and isolating <math>\theta</math> yields <math>\theta = y + \angle XDY + x</math>. Substituting <math>\theta</math> back into the first equation, we obtain |
<cmath>2x + 2y + \angle XCY + \angle XDY = 180^{\circ}</cmath> | <cmath>2x + 2y + \angle XCY + \angle XDY = 180^{\circ}</cmath> | ||
Since | Since | ||
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<cmath>37n \cdot 67n + 47^2 = 37 \cdot 67</cmath> | <cmath>37n \cdot 67n + 47^2 = 37 \cdot 67</cmath> | ||
<cmath>n^2 = \frac{270}{2479}</cmath> | <cmath>n^2 = \frac{270}{2479}</cmath> | ||
− | Now, since <math>\angle AYX = x</math> and <math>\angle BYX = y</math>, <math>\angle AYB = x + y</math>. From there, let <math>\angle AYD = \alpha</math> and <math>\angle BYC = \beta</math>. From angle chasing we can derive that <math>\angle YDX = \angle YAX = \beta - x</math> and <math>\angle YCX = \angle YBX = \alpha - y</math>. From there, since <math>\angle ADX = x</math>, it is quite clear that <math>\angle ADY = \beta</math>, and <math>\angle YAB = \beta</math> can be found similarly. From there, since <math>\angle ADY = \angle YAB = \angle BYC = \beta</math> and <math>\angle DAY = \angle AYB = \angle YBC = x + y</math>, we have <math>AA</math> similarity between <math>\triangle DAY</math>, <math>\triangle AYB</math>, and <math>\triangle YBC</math>. Therefore the length of <math>AY</math> is the geometric mean of the lengths of <math>DA</math> and <math>YB</math> (from <math>\triangle DAY \sim \triangle AYB</math>). However, <math>\triangle DAY \sim \triangle AYB \sim \triangle YBC</math> yields the proportion <math>\frac{AD}{DY} = \frac{YA}{AB} = \frac{BY}{YC}</math>; | + | Now, since <math>\angle AYX = x</math> and <math>\angle BYX = y</math>, <math>\angle AYB = x + y</math>. From there, let <math>\angle AYD = \alpha</math> and <math>\angle BYC = \beta</math>. From angle chasing we can derive that <math>\angle YDX = \angle YAX = \beta - x</math> and <math>\angle YCX = \angle YBX = \alpha - y</math>. From there, since <math>\angle ADX = x</math>, it is quite clear that <math>\angle ADY = \beta</math>, and <math>\angle YAB = \beta</math> can be found similarly. From there, since <math>\angle ADY = \angle YAB = \angle BYC = \beta</math> and <math>\angle DAY = \angle AYB = \angle YBC = x + y</math>, we have <math>AA</math> similarity between <math>\triangle DAY</math>, <math>\triangle AYB</math>, and <math>\triangle YBC</math>. Therefore the length of <math>AY</math> is the geometric mean of the lengths of <math>DA</math> and <math>YB</math> (from <math>\triangle DAY \sim \triangle AYB</math>). However, <math>\triangle DAY \sim \triangle AYB \sim \triangle YBC</math> yields the proportion <math>\frac{AD}{DY} = \frac{YA}{AB} = \frac{BY}{YC}</math>; hence, the length of <math>AB</math> is the geometric mean of the lengths of <math>DY</math> and <math>YC</math>. |
We can now simply use arithmetic to calculate <math>AB^2</math>. | We can now simply use arithmetic to calculate <math>AB^2</math>. | ||
<cmath>AB^2 = DY \cdot YC</cmath> | <cmath>AB^2 = DY \cdot YC</cmath> | ||
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'''-Solution by TheBoomBox77''' | '''-Solution by TheBoomBox77''' | ||
+ | ==Solution 5 (not too different)== | ||
+ | Let <math>E = DA \cap CB</math>. By Radical Axes, <math>E</math> lies on <math>XY</math>. Note that <math>EAXB</math> is cyclic as <math>X</math> is the Miquel point of <math>\triangle EDC</math> in this configuration. | ||
+ | |||
+ | Claim. <math>\triangle DXE \sim \triangle EXC</math> | ||
+ | Proof. We angle chase. <cmath>\measuredangle XEC = \measuredangle XEB = \measuredangle XAB = \measuredangle XDA = \measuredangle XDE</cmath>and<cmath>\measuredangle XCE = \measuredangle XCB = \measuredangle XBA = \measuredangle XEA = \measuredangle XED. \square</cmath> | ||
+ | |||
+ | Let <math>F = EX \cap AB</math>. Note <cmath>FA^2 = FX \cdot FY = FB^2</cmath>and<cmath>EF \cdot FX = AF \cdot FB = FA^2 = FX \cdot FY \implies EF = FY</cmath>By our claim, <cmath>\frac{DX}{XE} = \frac{EX}{XC} \implies EX^2 = DX \cdot XC = 67 \cdot 37 \implies FY = \frac{EY}{2} = \frac{EX+XY}{2} = \frac{\sqrt{67 \cdot 37}+47}{2}</cmath>and<cmath>FX=FY-47=\frac{\sqrt{67 \cdot 37}-47}{2}</cmath>Finally, <cmath>AB^2 = (2 \cdot FA)^2 = 4 \cdot FX \cdot FY = 4 \cdot \frac{(67 \cdot 37) - 47^2}{4} = \boxed{270}. \blacksquare</cmath>~Mathscienceclass | ||
+ | |||
+ | |||
+ | |||
+ | ==Solution 6 (No words)== | ||
+ | [[File:2016 AIME I 15.png|600px|right]] | ||
+ | [[File:2016 AIME I 15b.png|600px|left]] | ||
+ | |||
+ | <math>AB^2 = 4 AM^2 =2x(2x+ 2 XY) =(XP - XY) (XP + XY) = XP^2 - XY^2 = XC \cdot XD - XY^2 = 67 \cdot 37 - 47^2 = \boxed{270}.</math> | ||
+ | |||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/QoVIorvv_I8 | ||
+ | |||
+ | ~MathProblemSolvingSkills.com | ||
+ | |||
+ | ==Video Solution by The Power of Logic== | ||
+ | https://youtu.be/lTZx6tp2Fvg | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2016|n=I|num-b=14|after=Last Question}} | {{AIME box|year=2016|n=I|num-b=14|after=Last Question}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:29, 11 January 2023
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
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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