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Difference between revisions of "2022 AIME II Problems/Problem 15"

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==Solution==
 
==Solution==
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First observe that <math>AO_2 = O_2D</math> and <math>BO_1 = O_1C</math>. Let points <math>A'</math> and <math>B'</math> be the reflections of <math>A</math> and <math>B</math>, respectively, about the perpendicular bisector of <math>\overline{O_1O_2}</math>. Then quadrilaterals <math>ABO_1O_2</math> and <math>A'B'O_2O_1</math> are congruent, so hexagons <math>ABO_1CDO_2</math> and <math>B'A'O_1CDO_2</math> have the same area. Furthermore, triangles <math>DO_2B'</math> and <math>A'O_1C</math> are congruent, so <math>B'D = A'C</math> and quadrilateral <math>B'A'CD</math> is an isosceles trapezoid.
 +
<asy>
 +
import olympiad;
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size(180);
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defaultpen(linewidth(0.7));
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pair Bp = dir(105), Ap = dir(75), O1 = dir(25), C = dir(320), D = dir(220), O2 = dir(175);
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draw(unitcircle^^Bp--Ap--O1--C--D--O2--cycle);
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label("$B'$",Bp,dir(origin--Bp));
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label("$A'$",Ap,dir(origin--Ap));
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label("$O_1$",O1,dir(origin--O1));
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label("$C$",C,dir(origin--C));
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label("$D$",D,dir(origin--D));
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label("$O_2$",O2,dir(origin--O2));
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draw(O2--O1,linetype("4 4"));
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draw(Bp--D^^Ap--C,linetype("2 2"));
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</asy>
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Next, remark that <math>A'O_1 = DO_2</math>, so quadrilateral <math>A'O_1DO_2</math> is also an isosceles trapezoid; in turn, <math>A'D = O_1O_2 = 15</math>, and similarly <math>B'C = 15</math>. Thus, Ptolmey's theorem on <math>B'A'CD</math> yields <math>B'D\cdot A'C + 2\cdot 16 = 15^2</math>, whence <math>B'D = A'C = \sqrt{193}</math>. Let <math>\alpha = \angle B'A'D</math>. The Law of Cosines on triangle <math>B'A'D</math> yields
 +
<cmath>\cos\alpha = \frac{15^2 + 2^2 - (\sqrt{193})^2}{2\cdot 2\cdot 15} = \frac{36}{60} = \frac 35,</cmath> and hence <math>\sin\alpha = \tfrac 45</math>. Thus the distance between bases <math>AB</math> and <math>CD</math> is <math>12</math> (in fact, <math>\triangle B'A'D</math> is a <math>9-12-15</math> triangle with a <math>7-12-\sqrt{193}</math> triangle removed), which implies the area of <math>B'A'CD</math> is <math>\tfrac12\cdot 12\cdot(2+16) = 108</math>.
 +
 +
Now let <math>O_1C = O_2B' = r_1</math> and <math>O_2D = O_1A' = r_2</math>; the tangency of circles <math>\omega_1</math> and <math>\omega_2</math> implies <math>r_1 +  r_2 = 15</math>. Furthermore, angles <math>B'O_2D</math> and <math>B'A'D</math> are opposite angles in cyclic quadrilateral <math>A'B'O_2D</math>, which implies the measure of angle <math>B'O_2D</math> is <math>180^\circ - \alpha</math>. Therefore, the Law of Cosines applied to triangle <math>\triangle B'O_2D</math> yields
 +
<cmath>\begin{align*}
 +
193 &= r_1^2 + r_2^2 - 2r_1r_2(-\tfrac 35) = (r_1^2 + 2r_1r_2 + r_2^2) - \tfrac45r_1r_2\
 +
&= (r_1+r_2)^2 - \tfrac45 r_1r_2 = 225 - \tfrac45r_1r_2.
 +
\end{align*}</cmath>
 +
 +
Thus <math>r_1r_2 = 40</math>, and so the area of triangle <math>B'O_2D</math> is <math>\tfrac12r_1r_2\sin\alpha = 16</math>.
 +
 +
Thus, the area of hexagon <math>ABO_{1}CDO_{2}</math> is <math>108 + 2\cdot 16 = \boxed{140}</math>.
 +
 +
~djmathman
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=II|num-b=14|after=Last Problem}}
 
{{AIME box|year=2022|n=II|num-b=14|after=Last Problem}}
 +
 +
[[Category:Intermediate Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 14:34, 18 February 2022

Problem

Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("$\omega_1$",8*dir(110),SW); label("$\omega_2$",5*dir(70)+(15,0),SE); label("$O_1$",O1,W); label("$O_2$",O2,E); label("$B$",B,N+1/2*E); label("$A$",A,N+1/2*W); label("$C$",C,S+1/4*W); label("$D$",D,S+1/4*E); label("$15$",midpoint(O1--O2),N); label("$16$",midpoint(C--D),N); label("$2$",midpoint(A--B),S); label("$\Omega$",o.C+(o.r-1)*dir(270)); [/asy]

Solution

First observe that $AO_2 = O_2D$ and $BO_1 = O_1C$. Let points $A'$ and $B'$ be the reflections of $A$ and $B$, respectively, about the perpendicular bisector of $\overline{O_1O_2}$. Then quadrilaterals $ABO_1O_2$ and $A'B'O_2O_1$ are congruent, so hexagons $ABO_1CDO_2$ and $B'A'O_1CDO_2$ have the same area. Furthermore, triangles $DO_2B'$ and $A'O_1C$ are congruent, so $B'D = A'C$ and quadrilateral $B'A'CD$ is an isosceles trapezoid. [asy] import olympiad; size(180); defaultpen(linewidth(0.7)); pair Bp = dir(105), Ap = dir(75), O1 = dir(25), C = dir(320), D = dir(220), O2 = dir(175); draw(unitcircle^^Bp--Ap--O1--C--D--O2--cycle); label("$B'$",Bp,dir(origin--Bp)); label("$A'$",Ap,dir(origin--Ap)); label("$O_1$",O1,dir(origin--O1)); label("$C$",C,dir(origin--C)); label("$D$",D,dir(origin--D)); label("$O_2$",O2,dir(origin--O2)); draw(O2--O1,linetype("4 4")); draw(Bp--D^^Ap--C,linetype("2 2")); [/asy] Next, remark that $A'O_1 = DO_2$, so quadrilateral $A'O_1DO_2$ is also an isosceles trapezoid; in turn, $A'D = O_1O_2 = 15$, and similarly $B'C = 15$. Thus, Ptolmey's theorem on $B'A'CD$ yields $B'D\cdot A'C + 2\cdot 16 = 15^2$, whence $B'D = A'C = \sqrt{193}$. Let $\alpha = \angle B'A'D$. The Law of Cosines on triangle $B'A'D$ yields \[\cos\alpha = \frac{15^2 + 2^2 - (\sqrt{193})^2}{2\cdot 2\cdot 15} = \frac{36}{60} = \frac 35,\] and hence $\sin\alpha = \tfrac 45$. Thus the distance between bases $AB$ and $CD$ is $12$ (in fact, $\triangle B'A'D$ is a $9-12-15$ triangle with a $7-12-\sqrt{193}$ triangle removed), which implies the area of $B'A'CD$ is $\tfrac12\cdot 12\cdot(2+16) = 108$.

Now let $O_1C = O_2B' = r_1$ and $O_2D = O_1A' = r_2$; the tangency of circles $\omega_1$ and $\omega_2$ implies $r_1 + r_2 = 15$. Furthermore, angles $B'O_2D$ and $B'A'D$ are opposite angles in cyclic quadrilateral $A'B'O_2D$, which implies the measure of angle $B'O_2D$ is $180^\circ - \alpha$. Therefore, the Law of Cosines applied to triangle $\triangle B'O_2D$ yields \begin{align*} 193 &= r_1^2 + r_2^2 - 2r_1r_2(-\tfrac 35) = (r_1^2 + 2r_1r_2 + r_2^2) - \tfrac45r_1r_2\\ &= (r_1+r_2)^2 - \tfrac45 r_1r_2 = 225 - \tfrac45r_1r_2. \end{align*}

Thus $r_1r_2 = 40$, and so the area of triangle $B'O_2D$ is $\tfrac12r_1r_2\sin\alpha = 16$.

Thus, the area of hexagon $ABO_{1}CDO_{2}$ is $108 + 2\cdot 16 = \boxed{140}$.

~djmathman

See Also

2022 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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