Difference between revisions of "2020 AIME I Problems/Problem 15"

Latest revision as of 21:45, 27 November 2023

Problem

Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$

Solution 1

The following is a power of a point solution to this menace of a problem:

$[asy] defaultpen(fontsize(12)+0.6); size(250); pen p=fontsize(10)+gray+0.4; var phi=75.5, theta=130, r=4.8; pair A=r*dir(270-phi-57), C=r*dir(270+phi-57), B=r*dir(theta-57), H=extension(A,foot(A,B,C),B,foot(B,C,A)); path omega=circumcircle(A,B,C), c=circumcircle(H,B,C); pair O=circumcenter(H,B,C), Op=2*H-O, Z=bisectorpoint(O,Op), X=IP(omega,L(H,Z,0,50)), Y=IP(omega,L(H,Z,50,0)), D=2*H-A, K=extension(B,C,X,Y), E=extension(A,origin,X,Y), L=foot(H,B,C); draw(K--C^^omega^^c^^L(A,D,0,1)); draw(Y--K^^L(A,origin,0,1.5)^^L(X,D,0.7,3.5),fuchsia+0.4); draw(CR(H,length(H-L)),royalblue); draw(C--K,p); dot("$A$",A,dir(120)); dot("$B$",B,down); dot("$C$",C,down); dot("$H$",H,down); dot("$X$",X,up); dot("$Y$",Y,down); dot("$O$",origin,down); dot("$O'$",O,down); dot("$K$",K,up); dot("$L$",L,dir(H-X)); dot("$D$",D,down); dot("$E$",E,down); //draw(O--origin,p); //draw(origin--4*dir(57),fuchsia); //draw(Arc(H,3,160,200),royalblue); //draw(Arc(H,2,30,70),heavygreen); //draw(Arc(H,6,220,240),fuchsia); [/asy]$ Let points be what they appear as in the diagram below. Note that $3HX = HY$ is not insignificant; from here, we set $XH = HE = \frac{1}{2} EY = HL = 2$ by PoP and trivial construction. Now, $D$ is the reflection of $A$ over $H$ . Note $AO \perp XY$ , and therefore by Pythagorean theorem we have $AE = XD = \sqrt{5}$ . Consider $HD = 3$ . We have that $\triangle HXD \cong HLK$ , and therefore we are ready to PoP with respect to $(BHC)$ . Setting $BL = x, LC = y$ , we obtain $xy = 10$ by PoP on $(ABC)$ , and furthermore, we have $KH^2 = 9 = (KL - x)(KL + y) = (\sqrt{5} - x)(\sqrt{5} + y)$ . Now, we get $4 = \sqrt{5}(y - x) - xy$ , and from $xy = 10$ we take $\frac{14}{\sqrt{5}} = y - x.$ However, squaring and manipulating with $xy = 10$ yields that $(x + y)^2 = \frac{396}{5}$ and from here, since $AL = 5$ we get the area to be $3\sqrt{55} \implies \boxed{058}$ . ~awang11's sol

Solution 1a

As in the diagram, let ray $AH$ extended hits BC at L and the circumcircle at say $P$ . By power of the point at H, we have $HX \cdot HY = AH \cdot HP$ . The three values we are given tells us that $HP=\frac{2\cdot 6}{3}=4$ . L is the midpoint of $HP$ (see here: https://www.cut-the-knot.org/Curriculum/Geometry/AltitudeAndCircumcircle.shtml ), so $HL=LP=2$ .

As in the diagram provided, let K be the intersection of $BC$ and $XY$ . By power of a point on the circumcircle of triangle $HBC$ , $KH^{2}=KB \cdot KC$ . By power of a point on the circumcircle of triangle $ABC$ , $KB \cdot KC=KX \cdot KY$ , thus $KH^{2}=(KH-2)(KH+6)$ . Solving gives $4KH=12$ or $KH=3$ .

By the Pythagorean Theorem on triangle $HKL$ , $KL=\sqrt{5}$ . Now continue with solution 1.

Solution 2

$[asy] size(10cm); pair A, B, C, D, H, K, O, P, L, M, X, Y; A = (-15, 27); B = (-24, 0); C = (24, 0); D = (-8.28, 18.04); O = (0, 7); P = (0, -7); H = (-15, 13); K = (-15, -13); M = (0, 0); L = (-15, 0); X = (-24.9569, 5.53234); Y = (8.39688, 30.5477); draw(circle(O, 25)); draw(circle(P, 25)); draw(A--B--C--cycle); draw(H -- K); draw(A -- O -- P -- H -- cycle); draw(X -- Y); draw(O -- X, dashed); draw(O -- Y, dashed); draw(O -- B, dashed); draw(O -- C, dashed); label("$O$", O, ENE); label("$A$", A, NW); label("$B$", B, W); label("$C$", C, E); label("$H$", H, E); label("$H'$", K, NE); label("$X$", X, W); label("$Y$", Y, NE); label("$O'$", P, E); label("$M$", M, NE); label("$L$", L, NE); label("$D$", D, NNE); label("$2$", X -- H, NW); label("$3$", H -- A, SW); label("$6$", H -- Y, NW); label("$R$", O -- Y, E); dot(O); dot(P); dot(D); dot(H); [/asy]$ Diagram not to scale.

We first observe that $H'$ , the image of the reflection of $H$ over line $BC$ , lies on circle $O$ . This is because $\angle HBC = 90 - \angle C = \angle H'AC = \angle H'BC$ . This is a well known lemma. The result of this observation is that circle $O'$ , the circumcircle of $\triangle BHC$ is the image of circle $O$ over line $BC$ , which in turn implies that $\overline{AH} = \overline{OO'}$ and thus $AHO'O$ is a parallelogram. That $AHO'O$ is a parallelogram implies that $AO$ is perpendicular to $\overline{XY}$ , and thus divides segment $\overline{XY}$ in two equal pieces, $\overline{XD}$ and $\overline{DY}$ , of length $4$ .

Using Power of a Point, $\overline{AH} \cdot \overline{HH'} = \overline{XH} \cdot \overline{HY} \Longrightarrow 3 \cdot \overline{HH'} = 2 \cdot 6 \Longrightarrow \overline{HH'} = 4$ This means that $\overline{HL} = \frac12 \cdot 4 = 2$ and $\overline{AL} = 2 + 3 = 5$ , where $L$ is the foot of the altitude from $A$ onto $BC$ . All that remains to be found is the length of segment $\overline{BC}$ .

Looking at right triangle $\triangle AHD$ , we find that $\overline{AD} = \sqrt{\overline{AH}^2 - \overline{HD}^2} = \sqrt{3^2 - 2^2} = \sqrt{5}$ Looking at right triangle $\triangle ODY$ , we get the equation $\overline{OY}^2 - \overline{DY}^2 = \overline{OD}^2 = \left(\overline{AO} - \overline{AD}\right)^2$ Plugging in known values, and letting $R$ be the radius of the circle, we find that $R^2 - 16 = (R - \sqrt{5})^2 = R^2 - 2\sqrt5 R + 5 \Longrightarrow R = \frac{21\sqrt5}{10}$

Recall that $AHO'O$ is a parallelogram, so $\overline{AH} = \overline{OO'} = 3$ . So, $\overline{OM} = \frac32$ , where $M$ is the midpoint of $\overline{BC}$ . This means that $\overline{BC} = 2\overline{BM} = 2\sqrt{R^2 - \left(\frac32\right)^2} = 2\sqrt{\frac{441}{20} - \frac{9}{4}} = \frac{6\sqrt{55}}{5}$

Thus, the area of triangle $\triangle ABC$ is $\frac{\overline{AL} \cdot \overline{BC}}{2} = \frac{5 \cdot \frac{6\sqrt{55}}{5}}{2} = {3\sqrt{55}}$ The answer is $3 + 55 = \boxed{058}$ .

Solution 3 (Official MAA 1)

Extend $\overline{AH}$ to intersect $\omega$ again at $P$ . The Power of a Point Theorem yields $HP = \tfrac{HX \cdot HY}{HA} = 4$ . Because $\angle CAP=\angle CBP$ , and $\angle CAP$ and $\angle CBH$ are both complements to $\angle C$ , it follows that $\angle CBP = \angle CBH$ , implying that $\overline{BC}$ bisects $\overline{HP}$ , so the length of the altitude from $A$ to $\overline{BC}$ is $h_a = AH + \tfrac12 HP = 5$ .

Let the circumcircle of $\triangle BCH$ be $\omega'$ . Because $\triangle BCH \cong \triangle BCP$ , the two triangles must have the same circumradius. Because the circumcircle of $\triangle BCP$ is $\omega$ , the circles $\omega$ and $\omega'$ have the same radius $R$ . Denote the centers of $\omega$ and $\omega'$ by $O$ and $O'$ , respectively, and let $M$ be the midpoint of $\overline{XY}$ . Note that trapezoid $HMOO'$ has $\angle H = \angle M = 90^\circ$ . Also $HM = XM - XH = \frac12\cdot XY - HX = 2$ and $HO' = R$ . Because $\omega$ is a translation of $\omega'$ in the direction of $\overline{AH}$ , it follows that $OO' = AH = 3$ . Finally, the Pythagorean Theorem applied to $\triangle XMO$ yields $MO = \sqrt{R^2-16}$ . Let $T$ be the projection of $O$ onto $\overline{HO'}$ . Then $TO' = R-MO$ , so the Pythagorean Theorem applied to $\triangle TOO'$ yields $R - \sqrt{R^2-16} = \sqrt{3^2 - 2^2} = \sqrt{5}.$ Solving for $R$ gives $R = \tfrac{21}{2\sqrt5}$ . It follows from properties of the orthocenter $H$ that $\cos\angle A = \dfrac{AH}{2R} = \dfrac{\sqrt5}{7},$ so $\sin\angle A = \sqrt{1 - \cos^2\angle A} = \dfrac{2\sqrt{11}}{7}.$ Therefore by the Extended Law of Sines $a = BC = 2R \sin\angle A = \dfrac{6\sqrt{11}}{\sqrt5},$ so $[\triangle ABC] = \frac12 a h_a = \frac12 \cdot \frac{6\sqrt{11}}{\sqrt{5}} \cdot 5 = 3\sqrt{55}.$ The requested sum is $3+55 = 58$ .

$[asy] unitsize(0.6 cm); pair A, B, C, D, H, M, O, Op, P, T, X, Y, Z; real R = 21/(2*sqrt(5)); A = (7/sqrt(5),7/2); O = (0,0); B = intersectionpoint((0,-3/2)--(R,-3/2),Circle(O,R)); C = intersectionpoint((0,-3/2)--(-R,-3/2),Circle(O,R)); H = A + B + C; P = reflect(B,C)*(H); D = (H + P)/2; Op = reflect(B,C)*(O); X = intersectionpoint(H--(H + scale(2)*rotate(90)*(Op - H)), Circle(O,R)); Y = intersectionpoint(H--(H + scale(2)*rotate(90)*(H - Op)), Circle(O,R)); Z = extension(X, Y, B, C); M = (X + Y)/2; T = H + O - M; draw(Circle(O,R)); draw(Circle(Op,R)); draw(A--B--C--cycle); draw(A--P); draw(B--Z--Y); draw(H--Op--O--M); draw(O--T); draw(O--X); dot("$A$", A, NE); dot("$B$", B, SW); dot("$C$", C, W); dot("$D$", D, SW); dot("$H$", H, NE); dot("$M$", M, NE); dot("$O$", O, W); dot("$O'$", Op, W); dot("$P$", P, SE); dot("$T$", T, N); dot("$X$", X, E); dot("$Y$", Y, NW); dot("$Z$", Z, E); label("$\omega$", R*dir(140), dir(140)); label("$\omega'$", Op + R*dir(220), dir(220)); [/asy]$

Solution 3a (slightly modified Solution 3)

Note that $\overline{BC}$ bisects $\overline{OO'}$ . Using the same method from Solution 3 we find $R = \frac{21}{2\sqrt{5}}$ . Let the midpoint of $\overline{BC}$ be $N$ , then by the Pythagorean Theorem we have $BN^2 = OB^2 - ON^2 = R^2 - \frac{3}{2}^2$ , so $BN = \frac{3\sqrt{11}}{\sqrt{5}}$ . Since $h_a = 5$ we have that the area of ABC is $3\sqrt{55}$ so the answer is $3 + 55 = \boxed{58}$

~bobjoebilly

Solution 4 (Official MAA 2)

Let $D$ be the intersection point of line $AH$ and $\overline{BC}$ , noting that $\overline{AD}\perp\overline{BC}$ . Because the area of $\triangle ABC$ is $\tfrac12\cdot AD\cdot BC$ , it suffices to compute $AD$ and $BC$ separately. As in the previous solution, $AD = 5$ . The value of $BC$ can be found using the following lemma.

Lemma: Triangle $AXY$ is isosceles with base $\overline{XY}$ .

Proof: Because the circumcircle of $\triangle BCH$ , $\omega'$ , and $\omega$ have the same radius, there exists a translation $\Phi$ sending the former to the latter. Because $\overline{AH}$ is parallel to the line connecting the centers of the two circles, $\Phi$ must send $H$ to $A$ , meaning $\Phi$ also sends $\overline{XY}$ to the tangent to $\omega$ at $A$ . But this means that this tangent is parallel to $\overline{XY}$ , which implies the conclusion.

Applying Stewart's Theorem to $\triangle AXY$ yields $AX^2 = AH^2 + HX\cdot HY = 3^2 + 2\cdot 6 = 21,$ implying $AX = AY= \sqrt{21}.$

By the Law of Cosines $\cos \angle XAY = \frac{21 + 21 - 64}{2 \cdot 21} = -\frac{11}{21},$ so $\sin \angle XAY = \dfrac{4\sqrt{20}}{21}.$ Let $R$ be the radius of $\omega$ . By the Extended Law of Sines $R = \frac{XY}{2 \cdot \sin \angle XAY} = \dfrac{21}{\sqrt{20}}.$ Then the solution proceeds as in Solution 3.

Solution 5 (Official MAA 3)

Define points $D$ and $P$ as above, and note that $AD=5$ and $DH=PD=AD - AH = 2$ . Let the circumcircle of $\triangle BCH$ be $\omega'$ .

Extend $\overline{XY}$ past $X$ until it intersects line $BC$ at point $Z$ . Because line $BC$ is a radical axis of $\omega$ and $\omega'$ , it follows from the Power of a Point Theorem that $ZX \cdot ZY = ZX \cdot(ZX + 8) = ZH^2 = (ZX+2)^2,$ from which $ZX=1$ . By Pythagorean Theorem $ZD=\sqrt{ZH^2 - DH^2} = \sqrt{5}$ .

Let $m=CD$ and $n=BD$ . By the Power of a Point Theorem $mn= AD\cdot PD = 10.$ On the other hand, $ZH^2 = 9 = ZB \cdot ZC = (\sqrt{5} - n)(\sqrt{5} + m) = 5 + \sqrt{5}(m-n) - mn,$ from which $m-n = \frac{14}{\sqrt{5}}$ . Therefore

$(m+n)^2 = (m-n)^2 + 4mn = \frac{196}{5} + 40 = \frac{396}{5}.$ Thus $BC = m+n=\sqrt{\frac{396}{5}} = 6\sqrt{\frac{11}{5}}$ . Therefore $[\triangle ABC] = \frac{1}{2}BC \cdot AD = 3\sqrt{55},$ as above.

Solution 6

Let $O$ be circumcenter of $ABC,$ let $R$ be circumradius of $ABC,$ let $\omega'$ be the image of circle $\omega$ over line $BC$ (the circumcircle of $HBC$ ).

Let $P$ be the image of the reflection of $H$ over line $BC, P$ lies on circle $\omega.$ Let $M$ be the midpoint of $XY.$ Then $P$ lies on $\omega, OA = O'H, OA || O'H.$

(see here: https://brilliant.org/wiki/triangles-orthocenter/ or https://en.wikipedia.org/wiki/Altitude_(triangle) Russian)

$P$ lies on $\omega \implies OA = OP = R,$

$OA = O'H, OA || O'H \implies M$ lies on $OA.$

We use properties of crossing chords and get $AH \cdot HP = XH \cdot HY = 2 \cdot 6 \implies HP = 4, AP = AH + HP = 7.$ We use properties of radius perpendicular chord and get $MH = \frac{XH + HY}{2} – HY = 2.$ We find $\sin OAH =\frac{MH}{AH} = \frac{2}{3} \implies \cos OAH = \frac{\sqrt{5}}{3}.$ We use properties of isosceles $\triangle OAP$ and find $\hspace{5mm}R = \frac{AP}{2\cos OAP} = \frac{7}{2\frac {\sqrt{5}}{3}} = \frac{21}{2\sqrt{5}}.$

We use $OM' = \frac{AH}{2} = \frac {3}{2}$ and find $\hspace{25mm} \frac{BC}{2} = \sqrt{R^2 – OM'^2} = 3 \sqrt {\frac {11}{5}}.$

The area of $ABC$ $[ABC]=\frac{BC}{2} \cdot (AH + HD) = 3\cdot \sqrt{55} \implies 3+55 = \boldsymbol{\boxed{058}}.$ vladimir.shelomovskii@gmail.com, vvsss

Two Video Solutions by MOP 2024

https://youtube.com/watch?v=qgM-GbAWXCc (Power of a Point and Symmetry)
https://youtube.com/watch?v=XOUssYh14aw (Geometric Transformations, Angle Chasing, Trig)

~r00tsOfUnity

Video Solution by On the Spot STEM

https://www.youtube.com/watch?v=L7B20E95s4M

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