Difference between revisions of "2014 AIME I Problems/Problem 15"

Latest revision as of 21:30, 28 January 2024

Problem 15

In $\triangle ABC$ , $AB = 3$ , $BC = 4$ , and $CA = 5$ . Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$ , $\overline{BC}$ at $B$ and $D$ , and $\overline{AC}$ at $F$ and $G$ . Given that $EF=DF$ and $\frac{DG}{EG} = \frac{3}{4}$ , length $DE=\frac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$ .

Solution 1

Since $\angle DBE = 90^\circ$ , $DE$ is the diameter of $\omega$ . Then $\angle DFE=\angle DGE=90^\circ$ . But $DF=FE$ , so $\triangle DEF$ is a 45-45-90 triangle. Letting $DG=3x$ , we have that $EG=4x$ , $DE=5x$ , and $DF=EF=\frac{5x}{\sqrt{2}}$ .

Note that $\triangle DGE \sim \triangle ABC$ by SAS similarity, so $\angle BAC = \angle GDE$ and $\angle ACB = \angle DEG$ . Since $DEFG$ is a cyclic quadrilateral, $\angle BAC = \angle GDE=180^\circ-\angle EFG = \angle AFE$ and $\angle ACB = \angle DEG = \angle GFD$ , implying that $\triangle AFE$ and $\triangle CDF$ are isosceles. As a result, $AE=CD=\frac{5x}{\sqrt{2}}$ , so $BE=3-\frac{5x}{\sqrt{2}}$ and $BD =4-\frac{5x}{\sqrt{2}}$ .

Finally, using the Pythagorean Theorem on $\triangle BDE$ , $\left(3-\frac{5x}{\sqrt{2}}\right)^2 + \left(4-\frac{5x}{\sqrt{2}}\right)^2 = (5x)^2$ Solving for $x$ , we get that $x=\frac{5\sqrt{2}}{14}$ , so $DE= 5x =\frac{25 \sqrt{2}}{14}$ . Thus, the answer is $25+2+14=\boxed{041}$ .

Solution 2

$[asy] pair A = (0,3); pair B = (0,0); pair C = (4,0); draw(A--B--C--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); pair D = (2.21, 0); pair E = (0, 1.21); pair F = (1.71, 1.71); pair G = (2, 1.5); dot("$D$",D,dir(270)); dot("$E$",E,dir(180)); dot("$F$",F,dir(90)); dot("$G$",G,dir(0)); draw(Circle((1.109, 0.609), 1.28)); draw(D--E); draw(E--F); draw(D--F); draw(E--G); draw(D--G); draw(B--F); draw(B--G); [/asy]$

First we note that $\triangle DEF$ is an isosceles right triangle with hypotenuse $\overline{DE}$ the same as the diameter of $\omega$ . We also note that $\triangle DGE \sim \triangle ABC$ since $\angle EGD$ is a right angle and the ratios of the sides are $3:4:5$ .

From congruent arc intersections, we know that $\angle GED \cong \angle GBC$ , and that from similar triangles $\angle GED$ is also congruent to $\angle GCB$ . Thus, $\triangle BGC$ is an isosceles triangle with $BG = GC$ , so $G$ is the midpoint of $\overline{AC}$ and $AG = GC = 5/2$ . Similarly, we can find from angle chasing that $\angle ABF = \angle EDF = \frac{\pi}4$ . Therefore, $\overline{BF}$ is the angle bisector of $\angle B$ . From the angle bisector theorem, we have $\frac{AF}{AB} = \frac{CF}{CB}$ , so $AF = 15/7$ and $CF = 20/7$ .

Lastly, we apply power of a point from points $A$ and $C$ with respect to $\omega$ and have $AE \times AB=AF \times AG$ and $CD \times CB=CG \times CF$ , so we can compute that $EB = \frac{17}{14}$ and $DB = \frac{31}{14}$ . From the Pythagorean Theorem, we result in $DE = \frac{25 \sqrt{2}}{14}$ , so $a+b+c=25+2+14= \boxed{041}$

Also: $FG=\frac{20}{7}-\frac{5}{2}=\frac{5}{2}-\frac{15}{7}=\frac{5}{14}$ . We can also use Ptolemy's Theorem on quadrilateral $DEFG$ to figure what $FG$ is in terms of $d$ : $DE\cdot FG+DG\cdot EF=DF\cdot EG$ $d\cdot FG+\frac{3d}{5}\cdot \frac{d}{\sqrt{2}}=\frac{4d}{5}\cdot \frac{d}{\sqrt{2}}$ $d\cdot FG+\frac{3d^2}{5\sqrt{2}}=\frac{4d^2}{5\sqrt{2}}\implies FG=\frac{d}{5\sqrt{2}}$ Thus $\frac{d}{5\sqrt{2}}=\frac{5}{14}\rightarrow d=5\sqrt{2}\cdot\frac{5}{14}=\frac{25\sqrt{2}}{14}$ . $a+b+c=25+2+14= \boxed{041}$

Solution 3

Call $DE=x$ and as a result $DF=EF=\frac{x\sqrt{2}}{2}, EG=\frac{4x}{5}, GD=\frac{3x}{5}$ . Since $EFGD$ is cyclic we just need to get $DG$ and using LoS(for more detail see the $2$ nd paragraph of Solution $2$ ) we get $AG=\frac{5}{2}$ and using a similar argument(use LoS again) and subtracting you get $FG=\frac{5}{14}$ so you can use Ptolemy to get $x=\frac{25\sqrt{2}}{14} \implies \boxed{041}$ . ~First

Solution 4

See inside the $\triangle DEF$ , we can find that $AG>AF$ since if $AG<AF$ , we can see that Ptolemy Theorem inside cyclic quadrilateral $EFGD$ doesn't work. Now let's see when $AG>AF$ , since $\frac{DG}{EG} = \frac{3}{4}$ , we can assume that $EG=4x;GD=3x;ED=5x$ , since we know $EF=FD$ so $\triangle EFD$ is isosceles right triangle. We can denote $DF=EF=\frac{5x\sqrt{2}}{2}$ .Applying Ptolemy Theorem inside the cyclic quadrilateral $EFGD$ we can get the length of $FG$ can be represented as $\frac{x\sqrt{2}}{2}$ . After observing, we can see $\angle AFE=\angle EDG$ , whereas $\angle A=\angle EDG$ so we can see $\triangle AEF$ is isosceles triangle. Since $\triangle ABC$ is a $3-4-5$ triangle so we can directly know that the length of AF can be written in the form of $3x\sqrt{2}$ . Denoting a point $J$ on side $AC$ with that $DJ$ is perpendicular to side $AC$ . Now with the same reason, we can see that $\triangle DJG$ is a isosceles right triangle, so we can get $GJ=\frac{3x\sqrt{2}}{2}$ while the segment $CJ$ is $2x\sqrt{2}$ since its 3-4-5 again. Now adding all those segments together we can find that $AC=5=7x\sqrt{2}$ and $x=\frac{5\sqrt{2}}{14}$ and the desired $ED=5x=\frac{25\sqrt{2}}{14}$ which our answer is $\boxed{041}$ ~bluesoul

Solution 5

The main element of the solution is the proof that $BF$ is bisector of $\angle B.$

Let $O$ be the midpoint of $DE.$ $\angle EBF = 90^\circ \implies$

$O$ is the center of the circle $BDGFE.$ $\angle EOF = 90^\circ \implies \overset{\Large\frown} {EF} = 90^\circ \implies \angle EBF = 45^\circ \implies$ BF is bisector of $\angle ABC\implies BF = \frac {2AB \cdot BC}{AB+BC} \cos 45^\circ =\frac {12 \cdot \sqrt{2}}{7}.$ $\angle EGD = 90^\circ, \frac {EG}{GD}=\frac{4}{3} \implies$ $\angle GED = \angle GCD =\gamma \implies \overset{\Large\frown} {DG} = 2\gamma.$ $2\angle ACB = \overset{\Large\frown} {BEF} - \overset{\Large\frown} {DG} \implies \overset{\Large\frown} {BEF} = 4 \gamma \implies$ $\angle BOF = 4 \gamma \implies \angle OBF = \angle OFB = 90^\circ – 2 \gamma.$ Let $BO = EO = DO = r \implies BF = 2 r \cos(90^\circ – 2\gamma) =$ $=2 r \sin 2\gamma = 4r \sin \gamma \cdot \cos \gamma = 4 r\cdot \frac {3}{5} \cdot \frac {4}{5} = \frac {48}{25} = \frac {12 \cdot \sqrt{2}}{7}\implies$ $r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = 2r = \frac {25 \cdot \sqrt{2}}{14}\implies \boxed{\textbf{041}}.$ vladimir.shelomovskii@gmail.com, vvsss

Solution 6

The main element of the solution is the proof that $G$ is midpoint of $AC.$

As in Solution 5 we get $\angle GED = \angle DBG =\gamma \implies$

$\triangle BCG$ is isosceles triangle with $BG=CG.$

Similarly $BG = AG \implies AG = CG = BG = \frac {AC}{2} =\frac {5}{2}.$

$\overset{\Large\frown} {FG} = 90^\circ – \overset{\Large\frown} {GD} = 90^\circ – 2\gamma \implies$ $\overset{\Large\frown} {BFG} = 4\gamma + 90^\circ – 2\gamma = 90^\circ + 2\gamma \implies$ $\angle BOG = 90^\circ + 2\gamma \implies \angle BGO = \angle GBO = 45^\circ - \gamma.$ Let $\hspace{10mm} BO = EO = DO = r \implies$ $BG = 2 r \cos(45^\circ – \gamma) = 2 r (\sin \gamma + \cos \gamma)\frac {\sqrt {2}}{2} =$ $r \biggl(\frac {3}{5} + \frac {4}{5}\biggr) \sqrt {2} = r \frac {7 \sqrt{2}}{5} = \frac {5}{2}\implies$ $r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = \frac {25 \cdot \sqrt{2}}{14}\implies \boxed{\textbf{041}}.$ vladimir.shelomovskii@gmail.com, vvsss

Solution 7

Let $(BEFGD) = \omega$ . By Incenter-Excenter(Fact $5$ ), $F$ is the angle bisector of $\angle B$ . Then by Ratio Lemma we have $\frac{AG}{CG} = \frac{\sin(ABG)}{\sin(CBG)} \cdot \frac{AB}{BC} = \frac{\sin(GDE)}{\sin(DEG)} \cdot \frac{3}{4} = 1$ Thus, $G$ is the midpoint of $AC$ .

We can calculate $AF$ and $CF$ to be $\frac{15}{7}$ and $\frac{20}{7}$ respectively. And then by Power of a Point, we have $\newline$ $\operatorname{Pow}_{\omega}(A) = AE \cdot AB = AF \cdot AG \implies AE = \frac{25}{14}$ And then similarly, we have $CD = AE = \frac{25}{14}$ . $\newline$

Then $EB = \frac{17}{14}$ and $DB = \frac{31}{14}$ and by Pythagorean we have $DE = \frac{25\sqrt{2}}{14}$ , so our answer is $\boxed{\textbf{041}}.$

~dolphinday

Video Solution by mop 2024

https://youtu.be/GxxZYZrQl2A

~r00tsOfUnity

@@ Line 3: / Line 3: @@
 In <math>\triangle ABC</math>, <math>AB = 3</math>, <math>BC = 4</math>, and <math>CA = 5</math>. Circle <math>\omega</math> intersects <math>\overline{AB}</math> at <math>E</math> and <math>B</math>, <math>\overline{BC}</math> at <math>B</math> and <math>D</math>, and <math>\overline{AC}</math> at <math>F</math> and <math>G</math>. Given that <math>EF=DF</math> and <math>\frac{DG}{EG} = \frac{3}{4}</math>, length <math>DE=\frac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>.
-== Solution ==
+== Solution 1 ==
+Since <math>\angle DBE = 90^\circ</math>, <math>DE</math> is the diameter of <math>\omega</math>. Then <math>\angle DFE=\angle DGE=90^\circ</math>. But <math>DF=FE</math>, so <math>\triangle DEF</math> is a 45-45-90 triangle. Letting <math>DG=3x</math>, we have that <math>EG=4x</math>, <math>DE=5x</math>, and <math>DF=EF=\frac{5x}{\sqrt{2}}</math>.
+Note that <math>\triangle DGE \sim \triangle ABC</math> by SAS similarity, so <math>\angle BAC = \angle GDE</math> and <math>\angle ACB = \angle DEG</math>. Since <math>DEFG</math> is a cyclic quadrilateral, <math>\angle BAC = \angle GDE=180^\circ-\angle EFG = \angle AFE</math> and <math>\angle ACB = \angle DEG = \angle GFD</math>, implying that <math>\triangle AFE</math> and <math>\triangle CDF</math> are isosceles. As a result, <math>AE=CD=\frac{5x}{\sqrt{2}}</math>, so <math>BE=3-\frac{5x}{\sqrt{2}}</math> and <math>BD =4-\frac{5x}{\sqrt{2}}</math>.
+Finally, using the Pythagorean Theorem on <math>\triangle BDE</math>,
+<cmath> \left(3-\frac{5x}{\sqrt{2}}\right)^2 + \left(4-\frac{5x}{\sqrt{2}}\right)^2 = (5x)^2</cmath>
+Solving for <math>x</math>, we get that <math>x=\frac{5\sqrt{2}}{14}</math>, so <math>DE= 5x =\frac{25 \sqrt{2}}{14}</math>. Thus, the answer is <math>25+2+14=\boxed{041}</math>.
+== Solution 2 ==
 <asy>
@@ Line 40: / Line 50: @@
 Also: <math>FG=\frac{20}{7}-\frac{5}{2}=\frac{5}{2}-\frac{15}{7}=\frac{5}{14}</math>. We can also use Ptolemy's Theorem on quadrilateral <math>DEFG</math> to figure what <math>FG</math> is in terms of <math>d</math>:
-<cmath>DG\cdot EF+EF\cdot DG=DF\cdot EG</cmath>
+<cmath>DE\cdot FG+DG\cdot EF=DF\cdot EG</cmath>
-<cmath>\frac{3d}{5}\cdot \frac{d\sqrt{2}}{2}+d\cdot FG=\frac{4d}{5}\cdot \frac{d\sqrt{2}}{2}</cmath>
+<cmath>d\cdot FG+\frac{3d}{5}\cdot \frac{d}{\sqrt{2}}=\frac{4d}{5}\cdot \frac{d}{\sqrt{2}}</cmath>
-<cmath>\frac{3d\sqrt{2}}{10}+d\cdot EF=\frac{4d\sqrt{2}}{10}\implies FG=\frac{d\sqrt{2}}{10}</cmath>
+<cmath>d\cdot FG+\frac{3d^2}{5\sqrt{2}}=\frac{4d^2}{5\sqrt{2}}\implies FG=\frac{d}{5\sqrt{2}}</cmath>
-Thus <math>\frac{d\sqrt{2}}{10}=\frac{5}{14}\rightarrow d=5\sqrt{2}\cdot\frac{5}{14}=\frac{25\sqrt{2}}{14}</math>. <math>a+b+c=25+2+14= \boxed{041}</math>
+Thus <math>\frac{d}{5\sqrt{2}}=\frac{5}{14}\rightarrow d=5\sqrt{2}\cdot\frac{5}{14}=\frac{25\sqrt{2}}{14}</math>. <math>a+b+c=25+2+14= \boxed{041}</math>
+===Solution 3===
+Call <math>DE=x</math> and as a result <math>DF=EF=\frac{x\sqrt{2}}{2}, EG=\frac{4x}{5}, GD=\frac{3x}{5}</math>. Since <math>EFGD</math> is cyclic we just need to get <math>DG</math> and using LoS(for more detail see the <math>2</math>nd paragraph of Solution <math>2</math>) we get <math>AG=\frac{5}{2}</math> and using a similar argument(use LoS again) and subtracting you get <math>FG=\frac{5}{14}</math> so you can use Ptolemy to get <math>x=\frac{25\sqrt{2}}{14} \implies \boxed{041}</math>.
+~First
+==Solution 4==
+See inside the <math>\triangle DEF</math>, we can find that <math>AG>AF</math> since if <math>AG<AF</math>, we can see that Ptolemy Theorem inside cyclic quadrilateral <math>EFGD</math> doesn't work. Now let's see when <math>AG>AF</math>, since  <math>\frac{DG}{EG} = \frac{3}{4}</math>, we can assume that <math>EG=4x;GD=3x;ED=5x</math>, since we know <math>EF=FD</math> so <math>\triangle EFD </math> is isosceles right triangle. We can denote <math>DF=EF=\frac{5x\sqrt{2}}{2}</math>.Applying Ptolemy Theorem inside the cyclic quadrilateral <math>EFGD</math> we can get the length of <math>FG</math> can be represented as <math>\frac{x\sqrt{2}}{2}</math>. After observing, we can see <math>\angle AFE=\angle EDG</math>, whereas <math>\angle A=\angle EDG</math> so we can see <math>\triangle AEF</math> is isosceles triangle. Since <math>\triangle ABC</math> is a <math>3-4-5</math> triangle so we can directly know that the length of AF can be written in the form of <math>3x\sqrt{2}</math>. Denoting a point <math>J</math> on side <math>AC</math> with that <math>DJ</math> is perpendicular to side <math>AC</math>. Now with the same reason, we can see that <math>\triangle DJG</math> is a isosceles right triangle, so we can get <math>GJ=\frac{3x\sqrt{2}}{2}</math> while the segment <math>CJ</math> is <math>2x\sqrt{2}</math> since its 3-4-5 again. Now adding all those segments together we can find that <math>AC=5=7x\sqrt{2}</math> and <math>x=\frac{5\sqrt{2}}{14}</math> and the desired <math>ED=5x=\frac{25\sqrt{2}}{14}</math>
+which our answer is <math>\boxed{041}</math> ~bluesoul
+==Solution 5==
+[[File:2014 AIME II 15.png|450px|right]]
+The main element of the solution is the proof that <math>BF</math> is bisector of <math>\angle B.</math>
+Let <math>O</math> be the midpoint of <math>DE.</math> <math>\angle EBF = 90^\circ \implies</math>
+<math>O</math> is the center of the circle <math>BDGFE.</math>
+<math>\angle EOF = 90^\circ \implies \overset{\Large\frown} {EF} = 90^\circ \implies \angle EBF = 45^\circ \implies</math>
+BF is bisector of <math>\angle ABC\implies BF = \frac {2AB \cdot BC}{AB+BC} \cos 45^\circ =\frac {12 \cdot \sqrt{2}}{7}.</math>
+<cmath>\angle EGD = 90^\circ, \frac {EG}{GD}=\frac{4}{3} \implies</cmath>
+<cmath>\angle GED = \angle GCD =\gamma \implies  \overset{\Large\frown} {DG} = 2\gamma.</cmath>
+<cmath>2\angle ACB =  \overset{\Large\frown} {BEF} -  \overset{\Large\frown} {DG} \implies  \overset{\Large\frown} {BEF} = 4 \gamma \implies</cmath>
+<cmath>\angle BOF = 4 \gamma \implies \angle OBF = \angle OFB = 90^\circ – 2 \gamma.</cmath>
+Let <math>BO = EO = DO = r \implies BF = 2 r \cos(90^\circ – 2\gamma) =</math>
+<cmath>=2 r \sin 2\gamma = 4r \sin \gamma \cdot \cos \gamma = 4 r\cdot \frac {3}{5} \cdot \frac {4}{5} = \frac {48}{25} = \frac {12 \cdot \sqrt{2}}{7}\implies</cmath>
+<cmath>r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = 2r =  \frac {25 \cdot \sqrt{2}}{14}\implies  \boxed{\textbf{041}}.</cmath>
+'''vladimir.shelomovskii@gmail.com, vvsss'''
+==Solution 6 ==
+[[File:2014 AIME II 15a.png|450px|right]]
+The main element of the solution is the proof that <math>G</math> is midpoint of <math>AC.</math>
+As in Solution 5 we get <math>\angle GED = \angle DBG =\gamma \implies</math>
+<math>\triangle BCG </math> is isosceles triangle with <math>BG=CG.</math>
+Similarly <math>BG = AG \implies AG = CG = BG = \frac {AC}{2} =\frac {5}{2}.</math>
+<cmath>\overset{\Large\frown} {FG} = 90^\circ – \overset{\Large\frown} {GD} =  90^\circ – 2\gamma \implies</cmath>
+<cmath>\overset{\Large\frown} {BFG} = 4\gamma + 90^\circ – 2\gamma = 90^\circ + 2\gamma \implies</cmath>
+<cmath>\angle BOG = 90^\circ + 2\gamma \implies \angle BGO = \angle GBO = 45^\circ - \gamma.</cmath>
+Let <math>\hspace{10mm} BO = EO = DO = r \implies</math>
+<cmath>BG = 2 r \cos(45^\circ – \gamma) = 2 r (\sin \gamma + \cos \gamma)\frac {\sqrt {2}}{2} =</cmath>
+<cmath>r \biggl(\frac {3}{5} + \frac {4}{5}\biggr) \sqrt {2} = r \frac {7 \sqrt{2}}{5} = \frac {5}{2}\implies</cmath>
+<cmath>r = \frac {25 \cdot \sqrt{2}}{28}\implies ED = \frac {25 \cdot \sqrt{2}}{14}\implies  \boxed{\textbf{041}}.</cmath>
+'''vladimir.shelomovskii@gmail.com, vvsss'''
+==Solution 7==
+Let <math>(BEFGD) = \omega</math>.
+By Incenter-Excenter(Fact <math>5</math>), <math>F</math> is the angle bisector of <math>\angle B</math>.
+Then by Ratio Lemma we have
+<cmath>\frac{AG}{CG} = \frac{\sin(ABG)}{\sin(CBG)} \cdot \frac{AB}{BC} = \frac{\sin(GDE)}{\sin(DEG)} \cdot \frac{3}{4} = 1</cmath>
+Thus, <math>G</math> is the midpoint of <math>AC</math>.
+We can calculate <math>AF</math> and <math>CF</math> to be <math>\frac{15}{7}</math> and <math>\frac{20}{7}</math> respectively.
+And then by Power of a Point, we have
+<math>\newline</math>
+<cmath>\operatorname{Pow}_{\omega}(A) = AE \cdot AB = AF \cdot AG \implies AE = \frac{25}{14}</cmath>
+And then similarly, we have <math>CD = AE = \frac{25}{14}</math>.
+<math>\newline</math>
+Then <math>EB = \frac{17}{14}</math> and <math>DB = \frac{31}{14}</math> and by Pythagorean we have <math>DE = \frac{25\sqrt{2}}{14}</math>, so our answer is <math>\boxed{\textbf{041}}.</math>
+~dolphinday
+==Video Solution by mop 2024==
+https://youtu.be/GxxZYZrQl2A
+~r00tsOfUnity
 == See also ==
 {{AIME box|year=2014|n=I|num-b=14|after=Last Question}}
 {{MAA Notice}}

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