AIME 2016 I · 第 6 题

Solution 1

Suppose we label the angles as shown below.

As $\angle BCD$ and $\angle BAD$ intercept the same arc, we know that $\angle BAD=\gamma$. Similarly, $\angle ABD=\gamma$. Also, using $\triangle ICA$, we find $\angle CIA=180-\alpha-\gamma$. Therefore, $\angle AID=\alpha+\gamma=\angle DAI$, so $\triangle AID$ must be isosceles with $AD=ID=5$. Similarly, $BD=ID=5$. Then $\triangle DLB \sim \triangle ALC$, hence $\frac{AL}{AC} = \frac{3}{5}$. Also, $AI$ bisects $\angle LAC$, so by the Angle Bisector Theorem $\frac{CI}{IL} =\frac{AC}{AL}= \frac{5}{3}$. Thus $CI = \frac{10}{3}$, and the answer is $\boxed{013}$.

Solution 2 (Incenter/Excenter)

See the diagram in Solution 1.

Let $\Delta ABC$'s $C$-excenter be $J_C$. Since $CI$ is an angle bisector, $\angle ACD = \angle BCD$, meaning that $D$ is the midpoint of arc $\overarc{AB}$. By the Incenter/Excenter Lemma, $DA=DI=DB=DJ_C=2+3=5$, and applying Power of a Point on circle $D$ gives $AL\cdot LB = LI\cdot LJ_C = 2(3+5) = 16$. Applying Power of a Point again on $\Delta ABC$'s circumcircle gives $AL\cdot LB = LC\cdot LD = 16$, and since $LD = 3$, $LC = \frac{16}{3}$. Thus $IC = LC-LI = \frac{16}{3}-2 = \frac{10}{3}$. We submit $10+3=\boxed{013}$. - NamelyOrange

Solution 3

WLOG assume $\triangle ABC$ is isosceles. Then, $L$ is the midpoint of $AB$, and $\angle CLB=\angle CLA=90^\circ$. Draw the perpendicular from $I$ to $CB$, and let it meet $CB$ at $E$. Since $IL=2$, $IE$ is also $2$ (they are both inradii). Set $BD$ as $x$. Then, triangles $BLD$ and $CEI$ are similar, and $\tfrac{2}{3}=\tfrac{CI}{x}$. Thus, $CI=\tfrac{2x}{3}$. $\triangle CBD \sim \triangle CEI$, so $\tfrac{IE}{DB}=\tfrac{CI}{CD}$. Thus $\tfrac{2}{x}=\tfrac{(2x/3)}{(2x/3+5)}$. Solving for $x$, we have: $x^2-2x-15=0$, or $x=5, -3$. $x$ is positive, so $x=5$. As a result, $CI=\tfrac{2x}{3}=\tfrac{10}{3}$ and the answer is $\boxed{013}$

Solution 4

WLOG assume $\triangle ABC$ is isosceles (with vertex $C$). Let $O$ be the center of the circumcircle, $R$ the circumradius, and $r$ the inradius. A simple sketch will reveal that $\triangle ABC$ must be obtuse (as an acute triangle will result in $LI$ being greater than $DL$) and that $O$ and $I$ are collinear. Next, if $OI=d$, $DO+OI=R+d$ and $R+d=DL+LI=5$. Euler gives us that $d^{2}=R(R-2r)$, and in this case, $r=LI=2$. Thus, $d=\sqrt{R^{2}-4R}$. Solving for $d$, we have $R+\sqrt{R^{2}-4R}=5$, then $R^{2}-4R=25-10R+R^{2}$, yielding $R=\frac{25}{6}$. Next, $R+d=5$ so $d=\frac{5}{6}$. Finally, $OC=OI+IC$ gives us $R=d+IC$, and $IC=\frac{25}{6}-\frac{5}{6}=\frac{10}{3}$. Our answer is then $\boxed{013}$.

Solution 5

Since $\angle{LAD} = \angle{BDC}$ and $\angle{DLA}=\angle{DCB}$, $\triangle{DLA}\sim\triangle{DBC}$. Also, $\angle{DAC}=\angle{BLC}$ and $\angle{ACD}=\angle{LCB}$ so $\triangle{DAC}\sim\triangle{BLC}$. Now we can call $AC$, $b$ and $BC$, $a$. By angle bisector theorem, $\frac{AD}{DB}=\frac{AC}{BC}$. So let $AD=bk$ and $DB=ak$ for some value of $k$. Now call $IC=x$. By the similar triangles we found earlier, $\frac{3}{ak}=\frac{bk}{x+2}$ and $\frac{b}{x+5}=\frac{x+2}{a}$. We can simplify this to $abk^2=3x+6$ and $ab=(x+5)(x+2)$. So we can plug the $ab$ into the first equation and get $(x+5)(x+2)k^2=3(x+2) \rightarrow k^2(x+5)=3$. We can now draw a line through $A$ and $I$ that intersects $BC$ at $E$. By mass points, we can assign a mass of $a$ to $A$, $b$ to $B$, and $a+b$ to $D$. We can also assign a mass of $(a+b)k$ to $C$ by angle bisector theorem. So the ratio of $\frac{DI}{IC}=\frac{(a+b)k}{a+b}=k=\frac{2}{x}$. So since $k=\frac{2}{x}$, we can plug this back into the original equation to get $\left(\frac{2}{x}\right)^2(x+5)=3$. This means that $\frac{3x^2}{4}-x-5=0$ which has roots -2 and $\frac{10}{3}$ which means our $CI=\frac{10}{3}$ and our answer is $\boxed{013}$.

Solution 6

Since $\angle BCD$ and $\angle BAD$ both intercept arc $BD$, it follows that $\angle BAD=\gamma$. Note that $\angle AID=\alpha+\gamma$ by the external angle theorem. It follows that $\angle DAI=\angle AID=\alpha+\gamma$, so we must have that $\triangle AID$ is isosceles, yielding $AD=ID=5$. Note that $\triangle DLA \sim \triangle DAC$, so $\frac{DA}{DL} = \frac{DC}{DA}$. This yields $DC = \frac{25}{3}$. It follows that $CI = DC - DI = \frac{10}{3}$, giving a final answer of $\boxed{013}$.

Solution 7

Let $I_C$ be the excenter opposite to $C$ in $ABC$. By the incenter-excenter lemma $DI=DC \therefore$ $LI_C=8,LI=2,II_C=10$. Its well known that $(I_C,I,L,C)=-1 \implies \dfrac{LI_C}{LI}.\dfrac{CI}{CI_C}=-1 \implies \dfrac{CI}{CI+10}=\dfrac{1}{4} \implies \boxed{CI=\dfrac{10}{3}}$.$\blacksquare$ ~Pluto1708

Alternate solution: We can use the angle bisector theorem on $\triangle CBL$ and bisector $BI$ to get that $\tfrac{CI}{IL}=\tfrac{CI}{2}=\tfrac{BC}{BL}$. Since $\triangle CBL \sim \triangle ADL$, we get $\tfrac{BC}{BL}=\tfrac{AD}{DL}=\tfrac{5}{3}$. Thus, $CI=\tfrac{10}{3}$ and $p+q=\boxed{13}$. (https://artofproblemsolving.com/community/c759169h1918283_geometry_problem)

Solution 8

We can just say that quadrilateral $ADBC$ is a right kite with right angles at $A$ and $B$. Let us construct another similar right kite with the points of tangency on $AC$ and $BC$ called $E$ and $F$ respectively, point $I$, and point $C$. Note that we only have to look at one half of the circle since the diagram is symmetrical. Let us call $CI$ $x$ for simplicity's sake. Based on the fact that $\triangle BCD$ is similar to $\triangle FCI$ we can use triangle proportionality to say that $BD$ is $2\frac{x+5}{x}$. Using geometric mean theorem we can show that $BL$ must be $\sqrt{3x+6}$. With Pythagorean Theorem we can say that $3x+6+9=4{(\frac{x+5}{x})}^2$. Multiplying both sides by $x^2$ and moving everything to LHS will give you $3{x}^3+11{x}^2-40x-100=0$ Since $x$ must be in the form $\frac{p}{q}$ we can assume that $x$ is most likely a positive fraction in the form $\frac{p}{3}$ where $p$ is a factor of $100$. Testing the factors in synthetic division would lead $x = \frac{10}{3}$, giving us our desired answer $\boxed{013}$. ~Lopkiloinm

Solution 9 (Cyclic Quadrilaterals)

Connect $D$ to $A$ and $D$ to $B$ to form quadrilateral $ACBD$. Since quadrilateral $ACBD$ is cyclic, we can apply Ptolemy's Theorem on the quadrilateral.

Denote the length of $BD$ and $AD$ as $z$ (they must be congruent, as $\angle ABD$ and $\angle DAB$ are both inscribed in arcs that have the same degree measure due to the angle bisector intersecting the circumcircle at $D$), and the lengths of $BC$, $AC$, $AB$, and $CI$ as $a,b,c, x$, respectively.

After applying Ptolemy's, one will get that:

$$z(a+b)=c(x+5)$$ Next, since $ACBD$ is cyclic, triangles $ALD$ and $CLB$ are similar, yielding the following equation once simplifications are made to the equation $\frac{AD}{CB}=\frac{AL}{BL}$, with the length of $BL$ written in terms of $a,b,c$ using the angle bisector theorem on triangle $ABC$:

$$zc=3(a+b)$$ Next, drawing in the bisector of $\angle BAC$ to the incenter $I$, and applying the angle bisector theorem, we have that:

$$cx=2(a+b)$$ Now, solving for $z$ in the second equation, and $x$ in the third equation and plugging them both back into the first equation, and making the substitution $w=\frac{a+b}{c}$, we get the quadratic equation:

$$3w^2-2w-5=0$$ Solving, we get $w=5/3$, which gives $z=5$ and $x=10/3$, when we rewrite the above equations in terms of $w$. Thus, our answer is $\boxed{013}$ and we're done.

-mathislife52

Solution 10(Visual)

Solution 11

Let $AB=c,BC=a,CA=b$, and $x=\tfrac{a+b}{c}$. Then, notice that $\tfrac{CI}{IL}=\tfrac{a+b}{c}=x$, so $CI=IL\cdot{}x=2x$. Also, by the incenter-excenter lemma, $AD=BD=ID=IL+LD=5$. Therefore, by Ptolemy's Theorem on cyclic quadrilateral $ABCD$, $5a+5b=c(2x+5)$, so $5\left(\tfrac{a+b}{c}\right)=2x+5$, so $5x=2x+5$. Solving, we get that $x=\tfrac{5}{3}$, so $CI=\tfrac{10}{3}$ and the answer is $10+3=\boxed{013}$.

Solution 12

Perform a $\sqrt{bc}$ Inversion followed by a reflection along the angle bisector of $\angle BCA$.

It's well known that

$$AB \leftrightarrow \odot CBA \implies L \leftrightarrow D$$ $$I \leftrightarrow I_A$$ where $I_A$ is the $A-$excenter.

Also by Fact 5, $DI_A = 5$.

So,

$$CL \cdot CD = CI \cdot CI_A$$ $$\implies (CI + IL) \cdot (CI + ID) = (CI) \cdot (CI + II_A)$$ $$\implies (CI + 2) \cdot (CI + 5) = (CI) \cdot (CI + 10)$$ $$\implies 7CI +10= 10CI$$ $$\implies CI = \boxed{\dfrac{10}{3}}.\blacksquare$$ ~kamatadu

Solution 13

Without loss of generality, let $\triangle ABC$ be isosceles. Note that by the incenter-excenter lemma, $DI = DA = DB.$ Hence, $DA=DB=5.$ Let the point of tangency of the incircle and $\overline{BC}$ be $F$ and the point of tangency of the incircle and $\overline{AC}$ be $E.$ We note that $\angle ALC = \angle BLC = 90^\circ$ and $LA=LB=4,$ which immediately gives $AE=BF=4.$ Applying the Pythagorean Theorem on $\triangle ALC$ and $\triangle IEC$ gives $2^2+x^2=y^2$ and $4^2+(2+y)^2 = (4+x)^2.$ Solving for $y$ gives us $y=\frac{10}{3}.$ Therefore, $IC = \frac{10}{3}$ so the answer is $\boxed{13}.$

~peelybonehead

Solution 14 (Trig)

Let $C_1\in AB$ be the point such that $IC_1\perp AB$, and let $D_1\in AB$ be defined similarly for $D$. We know that $\triangle IC_1L\sim\triangle DD_1L$, so by triangle ratios $DD_1=\frac{3}{2}r$, where $r$ is the inradius. Additionally, by cyclic quadrilaterals, we know that $\angle BAD=\angle DAB=\frac{\gamma}{2}$, where $\gamma$ is equivalent to $\angle ACB$. Thus $\triangle ADB$ is isosceles and $DD_1$ is the perpendicular bisector of the triangle, so $AD_1=\frac{c}{2}$. Since $\tan\left(\frac{\gamma}{2}\right)=\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$ from formulas (where $s$ is half the perimeter of $\triangle ABC$) and since $\tan\left(\frac{\gamma}{2}\right)=\frac{\frac{3}{2}r}{\frac{1}{2}c}$ from $\triangle ADB$, we can set up an equation:

$\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}=\frac{\frac{3}{2}r}{\frac{1}{2}c}\implies\frac{\sqrt{s(s-a)(s-b)(s-c)}}{s(s-c)}=\frac{3rs}{cs}\implies\frac{[ABC]}{s(s-c)}=\frac{3[ABC]}{cs}$ $\implies\frac{1}{s-c}=\frac{3}{c}\implies \frac{s}{c}=\frac{4}{3}$

Let $C_2\in AB$ such that $CC_2\perp AB$. Then $CC_2=\frac{2[ABC]}{c}$. Using the area formula $[ABC]=rs$ and our fact from above yields $CC_2=\frac{8}{3}r$. We then notice that $\triangle CC_2L\sim\triangle IC_1L$, so if we let $x=CI$, by triangle ratios we find that $\frac{\frac{8}{3}r}{x+2}=\frac{r}{2}$, leading to $x=\frac{10}{3}$. Thus the answer is $10+3=\boxed{013}$.

~eevee9406