AIME 2013 II · 第 13 题

Solution 1

We can set $AE=ED=m$. Set $BD=k$, therefore $CD=3k, AC=4k$. Thereafter, by Stewart's Theorem on $\triangle ACD$ and cevian $CE$, we get $2m^2+14=25k^2$. Also apply Stewart's Theorem on $\triangle CEB$ with cevian $DE$. After simplification, $2m^2=17-6k^2$. Therefore, $k=1, m=\frac{\sqrt{22}}{2}$. Finally, note that (using [] for area) $[CED]=[CAE]=3[EDB]=3[AEB]=\frac{3}{8}[ABC]$, because of base-ratios. Using Heron's Formula on $\triangle EDB$, as it is simplest, we see that $[ABC]=3\sqrt{7}$, so your answer is $10$.

Solution 2

After drawing the figure, we suppose $BD=a$, so that $CD=3a$, $AC=4a$, and $AE=ED=b$.

Using Law of Cosines for $\triangle CED$ and $\triangle AEC$,we get

$$b^2+7-2b\sqrt{7}\cdot \cos(\angle CED)=9a^2\qquad (1)$$ $$b^2+7+2b\sqrt{7}\cdot \cos(\angle CED)=16a^2\qquad (2)$$ So, $(1)+(2)$, we get

$$2b^2+14=25a^2. \qquad (3)$$ Using Law of Cosines in $\triangle ACD$, we get

$$4b^2+9a^2-2\cdot 2b\cdot 3a\cdot \cos(\angle ADC)=16a^2$$ So,

$$\cos(\angle ADC)=\frac{4b^2-7a^2}{12ab}.\qquad (4)$$ Using Law of Cosines in $\triangle EDC$ and $\triangle EDB$, we get

$$b^2+9a^2-2\cdot 3a\cdot b\cdot \cos(\angle ADC)=7\qquad (5)$$ $$b^2+a^2+2\cdot a\cdot b\cdot \cos(\angle ADC)=9.\qquad (6)$$ $(5)+(6)$, and according to $(4)$, we can get

$$37a^2+2b^2=48. \qquad (7)$$ Using $(3)$ and $(7)$, we can solve $a=1$ and $b=\frac{\sqrt{22}}{2}$.

Finally, we use Law of Cosines for $\triangle ADB$,

$$4(\frac{\sqrt{22}}{2})^2+1+2\cdot2(\frac{\sqrt{22}}{2})\cdot \cos(ADC)=AB^2$$ then $AB=2\sqrt{7}$, so the height of this $\triangle ABC$ is $\sqrt{4^2-(\sqrt{7})^2}=3$.

Then the area of $\triangle ABC$ is $3\sqrt{7}$, so the answer is $\boxed{010}$.

Solution 3

Let $X$ be the foot of the altitude from $C$ with other points labelled as shown below.

Now we proceed using mass points. To balance along the segment $BC$, we assign $B$ a mass of $3$ and $C$ a mass of $1$. Therefore, $D$ has a mass of $4$. As $E$ is the midpoint of $AD$, we must assign $A$ a mass of $4$ as well. This gives $L$ a mass of $5$ and $M$ a mass of $7$.

Now let $AB=b$ be the base of the triangle, and let $CX=h$ be the height. Then as $AM:MB=3:4$, and as $AX=\frac{b}{2}$, we know that

$$MX=\frac{b}{2}-\frac{3b}{7}=\frac{b}{14}.$$ Also, as $CE:EM=7:1$, we know that $EM=\frac{1}{\sqrt{7}}$. Therefore, by the Pythagorean Theorem on $\triangle {XCM}$, we know that

$$\frac{b^2}{196}+h^2=\left(\sqrt{7}+\frac{1}{\sqrt{7}}\right)^2=\frac{64}{7}.$$ Also, as $LE:BE=5:3$, we know that $BL=\frac{8}{5}\cdot 3=\frac{24}{5}$. Furthermore, as $\triangle YLA\sim \triangle XCA$, and as $AL:LC=1:4$, we know that $LY=\frac{h}{5}$ and $AY=\frac{b}{10}$, so $YB=\frac{9b}{10}$. Therefore, by the Pythagorean Theorem on $\triangle BLY$, we get

$$\frac{81b^2}{100}+\frac{h^2}{25}=\frac{576}{25}.$$ Solving this system of equations yields $b=2\sqrt{7}$ and $h=3$. Therefore, the area of the triangle is $3\sqrt{7}$, giving us an answer of $\boxed{010}$.

Solution 4

Let the coordinates of $A$, $B$ and $C$ be $(-a, 0)$, $(a, 0)$ and $(0, h)$ respectively. Then $D = \Big(\frac{3a}{4}, \frac{h}{4}\Big)$ and $E = \Big(-\frac{a}{8},\frac{h}{8}\Big).$ $EC^2 = 7$ implies $a^2 + 49h^2 = 448$; $EB^2 = 9$ implies $81a^2 + h^2 = 576.$ Solve this system of equations simultaneously, $a=\sqrt{7}$ and $h=3$. Area of the triangle is $ah = 3\sqrt{7}$, giving us an answer of $\boxed{010}$.

Solution 5

Let $BD = x$. Then $CD = 3x$ and $AC = 4x$. Also, let $AE = ED = l$. Using Stewart's Theorem on $\bigtriangleup CEB$ gives us the equation $(x)(3x)(4x) + (4x)(l^2) = 27x + 7x$ or, after simplifying, $4l^2 = 34 - 12x^2$. We use Stewart's again on $\bigtriangleup CAD$: $(l)(l)(2l) + 7(2l) = (16x^2)(l) + (9x^2)(l)$, which becomes $2l^2 = 25x^2 - 14$. Substituting $2l^2 = 17 - 6x^2$, we see that $31x^2 = 31$, or $x = 1$. Then $l^2 = \frac{11}{2}$.

We now use Law of Cosines on $\bigtriangleup CAD$. $(2l)^2 = (4x)^2 + (3x)^2 - 2(4x)(3x)\cos C$. Plugging in for $x$ and $l$, $22 = 16 + 9 - 2(4)(3)\cos C$, so $\cos C = \frac{1}{8}$. Using the Pythagorean trig identity $\sin^2 + \cos^2 = 1$, $\sin^2 C = 1 - \frac{1}{64}$, so $\sin C = \frac{3\sqrt{7}}{8}$.

$[ABC] = \frac{1}{2} AC \cdot BC \sin C = (\frac{1}{2})(4)(4)(\frac{3\sqrt{7}}{8}) = 3\sqrt{7}$, and our answer is $3 + 7 = \boxed{010}$.

Note to writter: Couldn't we just use Heron's formula for $[CEB]$ after $x$ is solved then noticing that $[ABC] = 2 \times [CEB]$?

Solution 6 (Barycentric Coordinates)

Let ABC be the reference triangle, with $A=(1,0,0)$, $B=(0,1,0)$, and $C=(0,0,1)$. We can easily calculate $D=(0,\frac{3}{4},\frac{1}{4})$ and subsequently $E=(\frac{1}{2},\frac{3}{8},\frac{1}{8})$. Using distance formula on $\overline{EC}=(\frac{1}{2},\frac{3}{8},-\frac{7}{8})$ and $\overline{EB}=(\frac{1}{2},-\frac{5}{8},\frac{1}{8})$ gives

$$\begin{aligned} \begin{cases} 7&=|EC|^2=-a^2 \cdot \frac{3}{8} \cdot (-\frac{7}{8})-b^2 \cdot \frac{1}{2} \cdot (-\frac{7}{8})-c^2 \cdot \frac{1}{2} \cdot \frac{3}{8} \\ 9&=|EB|^2=-a^2 \cdot (-\frac{5}{8}) \cdot \frac{1}{8}-b^2 \cdot \frac{1}{2} \cdot \frac{1}{8}-c^2 \cdot \frac{1}{2} \cdot (-\frac{5}{8}) \\ \end{cases} \end{aligned}$$ But we know that $a=b$, so we can substitute and now we have two equations and two variables. So we can clear the denominators and prepare to cancel a variable:

$$\begin{aligned} \begin{cases} 7\cdot 64&=3\cdot 7\cdot a^2+b^2\cdot 4\cdot 7-c^2\cdot 4\cdot 3\\ 9\cdot 64&=5a^2-4b^2+4\cdot 5\cdot c^2 \\ \end{cases} \end{aligned}$$ $$\begin{aligned} \begin{cases} 7\cdot 64&=49a^2-12c^2 \\ 9\cdot 64&=a^2+20c^2 \\ \end{cases} \end{aligned}$$ $$\begin{aligned} \begin{cases} 5\cdot 7\cdot 64&=245a^2-60c^2 \\ 3\cdot 9\cdot 64&=3a^2+60c^2 \\ \end{cases} \end{aligned}$$ Then we add the equations to get

$$\begin{aligned} 62\cdot 64&=248a^2 \\ a^2 &=16 \\ a &=4 \\ \end{aligned}$$ Then plugging gives $b=4$ and $c=2\sqrt{7}$. Then the height from $C$ is $3$, and the area is $3\sqrt{7}$ and our answer is $\boxed{010}$.

Solution 7

Let $C=(0,0), A=(x,y),$ and $B=(-x,y)$. It is trivial to show that $D=\left(-\frac{3}{4}x,\frac{3}{4}y\right)$ and $E=\left(\frac{1}{8}x,\frac{7}{8}y\right)$. Thus, since $BE=3$ and $CE=\sqrt{7}$, we get that

$$\begin{aligned} \left(\frac{1}{8}x\right)^2+\left(\frac{7}{8}y\right)^2&=7 \\ \left(\frac{9}{8}x\right)^2+\left(\frac{1}{8}y\right)^2&=9 \\ \end{aligned}$$ Multiplying both equations by $64$, we get that

$$\begin{aligned} x^2+49y^2&=448 \\ 81x^2+y^2&=576 \\ \end{aligned}$$ Solving these equations, we get that $x=\sqrt{7}$ and $y=3$.

Thus, the area of $\triangle ABC$ is $xy=3\sqrt{7}$, so our answer is $\boxed{010}$.

Solution 8

The main in solution is to prove that $\angle BEC = 90^\circ$.

Let $M$ be midpoint $AB.$ Let $F$ be cross point of $AC$ and $BE.$

We use the formula for crossing segments in $\triangle ABC$ and get:

$$\frac {CF}{AF}= \frac {DE}{AE} \cdot (\frac {CD}{BD} + 1) = 1 \cdot (3 + 1) = 4.$$ $$\frac {FE }{BE}= \frac {CD}{BD} : (\frac {CF}{AF} + 1) = \frac {3}{5} \implies FE = \frac {9}{5}.$$ $$\triangle BCF:\hspace{5mm} BC = x, CF = \frac {4}{5}x, EF = \frac {9}{5}, BF = 3, CE = \sqrt{7}.$$ By Stewart's Theorem on $\triangle BCF$ and cevian $CE$, we get after simplification

$$x = 4 \implies BC^2 = CE^2 + BE^2 \implies \angle BEC = 90^\circ.$$ $$AE = ED, AM = MB \implies EM ||BC.$$ $\angle BEC = \angle CMB = 90^\circ \implies$ trapezium $BCEM$ is cyclic $\implies$

$$BM = CE, CM = BE \implies [ABC] = CM \cdot BM = 3 \sqrt {7} \implies 3+ 7 = \boxed{\textbf{010}}.$$ vladimir.shelomovskii@gmail.com, vvsss

Solution 9

Let $AB = 2x$ and let $y = BD.$ Then $CD = 3y$ and $AC = 4y.$

By the Law of Cosines on triangle $ABC,$

$$\cos C = \frac{16y^2 + 16y^2 - 4x^2}{2 \cdot 4y \cdot 4y} = \frac{32y^2 - 4x^2}{32y^2} = \frac{8y^2 - x^2}{8y^2}.$$ Then by the Law of Cosines on triangle $ACD,$ \begin{align*} AD^2 &= 16y^2 + 9y^2 - 2 \cdot 4y \cdot 3y \cdot \cos C \\ &= 25y^2 - 24y^2 \cdot \frac{8y^2 - x^2}{8y^2} \\ &= 3x^2 + y^2. \end{align*}Applying Stewart's Theorem to median $\overline{BE}$ in triangle $ABD,$ we get

$$BE^2 + AE \cdot DE = \frac{AB^2 + BD^2}{2}.$$ Thus,

$$9 + \frac{3x^2 + y^2}{4} = \frac{4x^2 + y^2}{2}.$$ This simplifies to $5x^2 + y^2 = 36.$

Applying Stewart's Theorem to median $\overline{CE}$ in triangle $ACD,$ we get

$$CE^2 + AE \cdot DE = \frac{AC^2 + CD^2}{2}.$$ Thus,

$$7 + \frac{3x^2 + y^2}{4} = \frac{16y^2 + 9y^2}{2}.$$ This simplifies to $3x^2 + 28 = 49y^2.$

Solving the system $5x^2 + y^2 = 36$ and $3x^2 + 28 = 49y^2,$ we find $x^2 = 7$ and $y^2 = 1,$ so $x = \sqrt{7}$ and $y = 1.$

Plugging this back in for our equation for $\cos C$ gives us $\frac{1}{8}$, so $\sin C = \frac{3\sqrt{7}}{8}.$ We can apply the alternative area of a triangle formula, where $AC \cdot BC \cdot \sin C \cdot \frac{1}{2} = 3\sqrt{7}.$ Therefore, our answer is $\boxed{010}$.

Video Solution

https://youtu.be/jVV4pYDGxGE?si=fDGGUOvCZRfdwUEz

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