2015 AIME II Problems/Problem 15

Problem

Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$ , respectively, and are externally tangent at point $A$ . Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$ . Points $B$ and $C$ lie on the same side of $\ell$ , and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

$[asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]$

Hint

This is a #15 on an AIME, so it must be difficult. Indeed, although there are several possible approaches, all of them are very computational.

Solution 1

$[asy] unitsize(35); draw(Circle((-1,0),1)); draw(Circle((4,0),4)); pair A,O_1, O_2, B,C,D,E,N,K,L,X,Y; A=(0,0);O_1=(-1,0);O_2=(4,0);B=(-24/15,-12/15);D=(-8/13,12/13);E=(32/13,-48/13);C=(24/15,-48/15);N=extension(E,B,O_2,O_1);K=foot(B,O_1,N);L=foot(C,O_2,N);X=foot(B,A,D);Y=foot(C,E,A); label("$A$",A,NE);label("$O_1$",O_1,NE);label("$O_2$",O_2,NE);label("$B$",B,SW);label("$C$",C,SW);label("$D$",D,NE);label("$E$",E,NE);label("$N$",N,W);label("$K$",(-24/15,0.2));label("$L$",(24/15,0.2));label("$n$",(-0.8,-0.12));label("$p$",((29/15,-48/15)));label("$\mathcal{P}$",(-1.6,1.1));label("$\mathcal{Q}$",(6,4)); draw(A--B--D--cycle);draw(A--E--C--cycle);draw(C--N);draw(O_2--N);draw(O_1--B,dashed);draw(O_2--C,dashed); dot(O_1);dot(O_2); draw(rightanglemark(O_1,B,N,5));draw(rightanglemark(O_2,C,N,5));draw(C--L,dashed);draw(B--K,dashed);draw(C--Y);draw(B--X); [/asy]$

Call $O_1$ and $O_2$ the centers of circles $\mathcal{P}$ and $\mathcal{Q}$ , respectively, and extend $CB$ and $O_2O_1$ to meet at point $N$ . Call $K$ and $L$ the feet of the altitudes from $B$ to $O_1N$ and $C$ to $O_2N$ , respectively. Using the fact that $\triangle{O_1BN} \sim \triangle{O_2CN}$ and setting $NO_1 = k$ , we have that $\frac{k+5}{k} = \frac{4}{1} \implies k=\frac{5}{3}$ . We can do some more length chasing using triangles similar to $OBN$ to get that $AK = AL = \frac{24}{15}$ , $BK = \frac{12}{15}$ , and $CL = \frac{48}{15}$ . Now, consider the circles $\mathcal{P}$ and $\mathcal{Q}$ on the coordinate plane, where $A$ is the origin. If the line $\ell$ through $A$ intersects $\mathcal{P}$ at $D$ and $\mathcal{Q}$ at $E$ then $4 \cdot DA = AE$ . To verify this, notice that $\triangle{AO_1D} \sim \triangle{EO_2A}$ from the fact that both triangles are isosceles with $\angle{O_1AD} \cong \angle{O_2AE}$ , which are corresponding angles. Since $O_2A = 4\cdot O_1A$ , we can conclude that $4 \cdot DA = AE$ .

Hence, we need to find the slope $m$ of line $\ell$ such that the perpendicular distance $n$ from $B$ to $AD$ is four times the perpendicular distance $p$ from $C$ to $AE$ . This will mean that the product of the bases and heights of triangles $ACE$ and $DBA$ will be equal, which in turn means that their areas will be equal. Let the line $\ell$ have the equation $y = -mx \implies mx + y = 0$ , and let $m$ be a positive real number so that the negative slope of $\ell$ is preserved. Setting $A = (0,0)$ , the coordinates of $B$ are $(x_B, y_B) = \left(\frac{-24}{15}, \frac{-12}{15}\right)$ , and the coordinates of $C$ are $(x_C, y_C) = \left(\frac{24}{15}, \frac{-48}{15}\right)$ . Using the point-to-line distance formula and the condition $n = 4p$ , we have $\frac{|mx_B + 1(y_B) + 0|}{\sqrt{m^2 + 1}} = \frac{4|mx_C + 1(y_C) + 0|}{\sqrt{m^2 + 1}}$ $\implies |mx_B + y_B| = 4|mx_C + y_C| \implies \left|\frac{-24m}{15} + \frac{-12}{15}\right| = 4\left|\frac{24m}{15} + \frac{-48}{15}\right|.$ If $m > 2$ , then clearly $B$ and $C$ would not lie on the same side of $\ell$ . Thus since $m > 0$ , we must switch the signs of all terms in this equation when we get rid of the absolute value signs. We then have $\frac{6m}{15} + \frac{3}{15} = \frac{48}{15} - \frac{24m}{15} \implies 2m = 3 \implies m = \frac{3}{2}.$ Thus, the equation of $\ell$ is $y = -\frac{3}{2}x$ .

Then we can find the coordinates of $D$ by finding the point $(x,y)$ other than $A = (0,0)$ where the circle $\mathcal{P}$ intersects $\ell$ . $\mathcal{P}$ can be represented with the equation $(x + 1)^2 + y^2 = 1$ , and substituting $y = -\frac{3}{2}x$ into this equation yields $x = 0, -\frac{8}{13}$ as solutions. Discarding $x = 0$ , the $y$ -coordinate of $D$ is $-\frac{3}{2} \cdot -\frac{8}{13} = \frac{12}{13}$ . The distance from $D$ to $A$ is then $\frac{4}{\sqrt{13}}.$ The perpendicular distance from $B$ to $AD$ or the height of $\triangle{DBA}$ is $\frac{|\frac{3}{2}\cdot\frac{-24}{15} + \frac{-12}{15} + 0|}{\sqrt{\frac{3}{2}^2 + 1}} = \frac{\frac{48}{15}}{\frac{\sqrt{13}}{2}} = \frac{32}{5\sqrt{13}}.$ Finally, the common area is $\frac{1}{2}\left(\frac{32}{5\sqrt{13}} \cdot \frac{4}{\sqrt{13}}\right) = \frac{64}{65}$ , and $m + n = 64 + 65 = \boxed{129}$ .

Solution 2

By homothety, we deduce that $AE = 4 AD$ . (The proof can also be executed by similar triangles formed from dropping perpendiculars from the centers of $P$ and $Q$ to $l$ .) Therefore, our equality of area condition, or the equality of base times height condition, reduces to the fact that the distance from $B$ to $l$ is four times that from $C$ to $l$ . Let the distance from $C$ be $x$ and the distance from $B$ be $4x$ .

Let $P$ and $Q$ be the centers of their respective circles. Then dropping a perpendicular from $P$ to $Q$ creates a $3-4-5$ right triangle, from which $BC = 4$ and, if $\alpha = \angle{AQC}$ , that $\cos \alpha = \dfrac{3}{5}$ . Then $\angle{BPA} = 180^\circ - \alpha$ , and the Law of Cosines on triangles $APB$ and $AQC$ gives $AB = \dfrac{4}{\sqrt{5}}$ and $AC = \dfrac{8}{\sqrt{5}}.$

Now, using the Pythagorean Theorem to express the length of the projection of $BC$ onto line $l$ gives $\sqrt{\frac{16}{5} - 16x^2} + \sqrt{\frac{64}{5} - x^2} = \sqrt{16 - 9x^2}.$ Squaring and simplifying gives $\sqrt{\left(\frac{1}{5} - x^2\right)\left(\frac{64}{5} - x^2\right)} = x^2,$ and squaring and solving gives $x = \dfrac{8}{5\sqrt{13}}.$

By the Law of Sines on triangle $ABD$ , we have $\frac{BD}{\sin A} = 2.$ But we know $\sin A = \dfrac{4x}{AB}$ , and so a small computation gives $BD = \dfrac{16}{\sqrt{65}}.$ The Pythagorean Theorem now gives $AD = \sqrt{BD^2 - (4x)^2} + \sqrt{AB^2 - (4x)^2} = \frac{4}{\sqrt{13}},$ and so the common area is $\dfrac{1}{2} \cdot \frac{4}{\sqrt{13}} \cdot \frac{32}{5\sqrt{13}} = \frac{64}{65}.$ The answer is $\boxed{129}.$

Alternate Path to x

Call the intersection of lines $l$ and $BC$ $E$ .You can use similar triangles to find that the distance from $B$ to $E$ is four times the distance from $C$ to $E$ . Then draw a perpendicular from $A$ to $BC$ and call the point $F$ . $AF = \frac{8}{5}$ and $FE = FC + CE = \frac{16}{5} + \frac{4}{3} = \frac{68}{15}$ , so by the Pythagorean Theorem, $AE = \dfrac{20\sqrt{13}}{5}$ . You can now use similar triangles to find that $x = \dfrac{8}{5\sqrt{13}}$ and continue on like in solution 2.

Solution 3

$DE$ goes through $A$ , the point of tangency of both circles. So $DE$ intercepts equal arcs in circle $P$ and $Q$ : homothety. Hence, $AE=4AD$ . We will use such similarity later.

The diagonal distance between the centers of the circles is $4+1=5$ . The difference in heights is $4-1=3$ . So $BC=\sqrt{5^2-3^2}=4$ .

The triangle connecting the centers with a side parallel to $BC$ is a $3-4-5$ right triangle. Since $O_PA=1$ , the height of $A$ is $1+3/5=8/5$ . Drop an altitude from $A$ to $BC$ and call it $I$ : $IB=4/5$ and $IC=4-4/5=32/5$ . Since right $\triangle AIB\sim\triangle CIB$ , $ABC$ is a right triangle also; $IB:IA:IC$ form a geometric progression $\times 2$ .

Extend $BA$ through $A$ to a point $G$ on the other side of $\circ Q$ . By homothety, $\triangle DAB\sim\triangle EAG$ . By angle chasing $\triangle DAB$ through right triangle $ABC$ , we deduce that $\angle CEG$ is a right angle. Since $ACEG$ is cyclic, $\angle GAC$ is also right. So $CG$ is a diameter of $\circ G$ . Because of this, $CG \perp BC$ , the tangent line. $\triangle BCG$ is right and $\triangle BCG\sim\triangle ABC\sim\triangle CAG$ .

$AC=\sqrt{(8/5)^2+(16/5)^2}=8\sqrt{5}/5$ so $AG=2AC=16\sqrt{5}/5$ and $[\triangle CAG]=64/5$ .

Since $[\triangle DAB]=[\triangle ACE]$ , the common area is $[ACEG]/17$ . $16[\triangle DAB]=[\triangle GAE]$ because the triangles are similar with a ratio of $1:4$ . So we only need to find $[\triangle CEG]$ now.

Extend $DE$ through $E$ to intersect the tangent at $F$ . Because $4DA=AE$ , the altitude from $B$ to $AD$ is $4$ times the height from $C$ to $EA$ . So $BC=3/4BF$ and $BF=16/3$ . We look at right triangle $\triangle AIF$ . $IF=68/15$ and $AI=8/5$ . $\triangle AIF$ is a $17-6-5\sqrt{13}$ right triangle. Hypotenuse $AF$ intersects $CG$ at a point, we call it $H$ . $CH=4/3\div 68/15\cdot 8/5=8/17$ . So $HG=8-8/17=128/17$ .

By Power of a Point, $CH\cdot HG=AH\cdot HE$ . $AH=16/5\cdot 5\sqrt{13}/17=16\sqrt{13}/17.$ So $HE=1024/289\cdot 17/(16\sqrt{13})=64/(17\sqrt{13})$ . The height from $E$ to $CG$ is $17/(5\sqrt{13})\cdot 64/(17\sqrt{13})=64/65$ .

Thus, $[\triangle CEG]=64/65\cdot 8\div 2=256/65$ . The area of the whole cyclic quadrilateral is $64/5+256/65=(832+256)/65=1088/65$ . Lastly, the common area is $1/17$ the area of the quadrilateral, or $64/65$ . So $64+65=\boxed{129}$ .

Solution 4

Let the center of the smaller circle be $O_1$ , the center of the larger be $O_2$ , and the origin be $B$ . We find that the point of tangency is $(\frac{4}{5}, \frac{8}{5})$ and $C$ is $(4,0)$ by drawing a horizontal from $O_1$ to $O_2 C$ .

Let $D=(a,b)$ and $E=(c,d)$ . Then, using Shoelace we have: $\frac{4b-8a}{10}=\frac{-32-4d+8c+20d}{10},$ or $2a-b+2c+4d=8.$

Using Homothety, we also know that $4(4/5-a)=c-4/5$ and $4(b-8/5)=8/5-d.$ We rearrange and plug into our area equation from above to get $6a+17b=32.$ Plug this into the equation for Circle 1, $a^2+(b-1)^2=1$ to get that $a=-\frac{16}{65}, b=\frac{128}{65}.$ Use our area formula from above to get that the area is $\frac{64}{65},$ so our answer is $64+65=\boxed{129}.$

2015 AIME II (Problems • Answer Key • Resources)
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