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Difference between revisions of "1994 USAMO Problems/Problem 3"

(Created page with "==Problem== A convex hexagon <math>ABCDEF</math> is inscribed in a circle such that <math>AB=CD=EF</math> and diagonals <math>AD,BE</math>, and <math>CF</math> are concurrent. Le...")
 
(→‎Solution)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
 +
 +
Let the diagonals <math>AD</math>, <math>BE</math>, <math>CF</math> meet at <math>Q</math>.
 +
 +
First, let's show that the triangles <math>\triangle AEC</math> and <math>\triangle QED</math> are similar.
 +
 +
<center>
 +
<asy>
 +
 +
pair A,B,C,D,E,F,P,Q;
 +
A=(-0.96,0.28);
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B=(-0.352,0.936);
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C=(0,1);
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D=(4/5,3/5);
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E=(4/5,-3/5);
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F=(0,-1);
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P=IntersectionPoint(A--D,C--E);
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Q=IntersectionPoint(A--D,C--F);
 +
 +
draw(A--B);
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draw(B--C);
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draw(C--D);
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draw(D--E,green);
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draw(E--F);
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draw(F--A);
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draw(A--C,red);
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draw(A--Q);
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draw(A--E,red);
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draw(B--Q);
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draw(C--E,red);
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draw(C--F);
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draw(Q--E,green);
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draw(Q--D,green);
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draw(circle((0,0),1));
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label("\(A\)",A,W);
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label("\(B\)",B,N);
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label("\(C\)",C,N);
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label("\(D\)",D,NE);
 +
label("\(E\)",E,SE);
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label("\(F\)",F,S);
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label("\(P\)",(0.3,0.8),S);
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label("\(Q\)",(-0.15,0.4),S);
 +
 +
</asy>
 +
</center>
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<math>\angle ACE=\angle ADE</math> because <math>A</math>,<math>C</math>,<math>D</math> and <math>E</math> all lie on the circle, and <math>\angle ADE=\angle QDE</math>. <math>\angle AEB=\angle CED</math> because <math>AB=CD</math>, and <math>A</math>,<math>B</math>,<math>C</math>,<math>D</math> and <math>E</math> all lie on the circle. Then,
 +
 +
<math>\angle AEB=\angle CED \rightarrow \angle AEB+\angle BEC=\angle CED+\angle BEC \rightarrow \angle AEC=\angle QED</math>
 +
 +
Therefore, <math>\triangle AEC</math> and <math>\triangle QED</math> are similar, so <math>AC/CE=QD/DE</math>.
 +
 +
Next, let's show that <math>\triangle AEC</math> and <math>\triangle CDQ</math> are similar.
 +
 +
<center>
 +
<asy>
 +
 +
pair A,B,C,D,E,F,P,Q;
 +
A=(-0.96,0.28);
 +
B=(-0.352,0.936);
 +
C=(0,1);
 +
D=(4/5,3/5);
 +
E=(4/5,-3/5);
 +
F=(0,-1);
 +
P=IntersectionPoint(A--D,C--E);
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Q=IntersectionPoint(A--D,C--F);
 +
 +
draw(A--B);
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draw(B--C);
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draw(C--D,green);
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draw(D--E);
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draw(E--F);
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draw(F--A);
 +
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draw(A--C,red);
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draw(A--Q);
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draw(A--E,red);
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draw(B--E);
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draw(C--E,red);
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draw(Q--F);
 +
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draw(Q--C,green);
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draw(Q--D,green);
 +
 +
draw(circle((0,0),1));
 +
 +
label("\(A\)",A,W);
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label("\(B\)",B,N);
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label("\(C\)",C,N);
 +
label("\(D\)",D,NE);
 +
label("\(E\)",E,SE);
 +
label("\(F\)",F,S);
 +
label("\(P\)",(0.3,0.8),S);
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label("\(Q\)",(-0.15,0.4),S);
 +
 +
</asy>
 +
</center>
 +
 +
<math>\angle AEC=\angle ADC</math> because <math>A</math>,<math>C</math>,<math>D</math> and <math>E</math> all lie on the circle, and <math>\angle ADC=\angle CDQ</math>. <math>\angle EAD=\angle ECD</math> because <math>A</math>,<math>C</math>,<math>D</math> and <math>E</math> all lie on the circle. <math>\angle DAC=\angle ECF</math> because <math>CD=EF</math>, and <math>A</math>,<math>C</math>,<math>D</math>,<math>E</math> and <math>F</math> all lie on the circle. Then,
 +
 +
<math>\angle EAC=\angle EAD+\angle DAC=\angle ECD+\angle ECF=\angle DCY</math>
 +
 +
Therefore, <math>\triangle AEC</math> and <math>\triangle CDQ</math> are similar, so <math>AC/CE=CQ/QD</math>.
 +
 +
Lastly, let's show that <math>\triangle CPQ</math> and <math>\triangle EPD</math> are similar.
 +
 +
<center>
 +
<asy>
 +
 +
pair A,B,C,D,E,F,P,Q;
 +
A=(-0.96,0.28);
 +
B=(-0.352,0.936);
 +
C=(0,1);
 +
D=(4/5,3/5);
 +
E=(4/5,-3/5);
 +
F=(0,-1);
 +
P=IntersectionPoint(A--D,C--E);
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Q=IntersectionPoint(A--D,C--F);
 +
 +
draw(A--B);
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draw(B--C);
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draw(C--D);
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draw(D--E,green);
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draw(E--F);
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draw(F--A);
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draw(A--C);
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draw(A--Q);
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draw(A--E);
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draw(B--E);
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draw(P--E,green);
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draw(Q--F);
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draw(C--P,red);
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draw(Q--P,red);
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draw(C--Q,red);
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draw(D--P,green);
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draw(circle((0,0),1));
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label("\(A\)",A,W);
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label("\(B\)",B,N);
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label("\(C\)",C,N);
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label("\(D\)",D,NE);
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label("\(E\)",E,SE);
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label("\(F\)",F,S);
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label("\(P\)",(0.3,0.8),S);
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label("\(Q\)",(-0.15,0.4),S);
 +
 +
</asy>
 +
</center>
 +
 +
Because <math>CD=EF</math> and <math>C</math>,<math>D</math>,<math>E</math> and <math>F</math> all lie on the circle, <math>CF</math> is parallel to <math>DE</math>. So, <math>\triangle CPQ</math> and <math>\triangle EPD</math> are similar, and <math>CQ/DE=CP/PE</math>.
 +
 +
Putting it all together, <math>CP/PE=CQ/DE=AC/CE\cdot QD/DE=(AC/CE)^2</math>.

Revision as of 16:34, 30 May 2014

Problem

A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\frac{CP}{CE}=(\frac{AC}{CE})^2$.

Solution

Let the diagonals $AD$, $BE$, $CF$ meet at $Q$.

First, let's show that the triangles $\triangle AEC$ and $\triangle QED$ are similar.

[asy] pair A,B,C,D,E,F,P,Q; A=(-0.96,0.28); B=(-0.352,0.936); C=(0,1); D=(4/5,3/5); E=(4/5,-3/5); F=(0,-1); P=IntersectionPoint(A--D,C--E); Q=IntersectionPoint(A--D,C--F); draw(A--B); draw(B--C); draw(C--D); draw(D--E,green); draw(E--F); draw(F--A); draw(A--C,red); draw(A--Q); draw(A--E,red); draw(B--Q); draw(C--E,red); draw(C--F); draw(Q--E,green); draw(Q--D,green); draw(circle((0,0),1)); label("\(A\)",A,W); label("\(B\)",B,N); label("\(C\)",C,N); label("\(D\)",D,NE); label("\(E\)",E,SE); label("\(F\)",F,S); label("\(P\)",(0.3,0.8),S); label("\(Q\)",(-0.15,0.4),S); [/asy]

$\angle ACE=\angle ADE$ because $A$,$C$,$D$ and $E$ all lie on the circle, and $\angle ADE=\angle QDE$. $\angle AEB=\angle CED$ because $AB=CD$, and $A$,$B$,$C$,$D$ and $E$ all lie on the circle. Then,

$\angle AEB=\angle CED \rightarrow \angle AEB+\angle BEC=\angle CED+\angle BEC \rightarrow \angle AEC=\angle QED$

Therefore, $\triangle AEC$ and $\triangle QED$ are similar, so $AC/CE=QD/DE$.

Next, let's show that $\triangle AEC$ and $\triangle CDQ$ are similar.

[asy] pair A,B,C,D,E,F,P,Q; A=(-0.96,0.28); B=(-0.352,0.936); C=(0,1); D=(4/5,3/5); E=(4/5,-3/5); F=(0,-1); P=IntersectionPoint(A--D,C--E); Q=IntersectionPoint(A--D,C--F); draw(A--B); draw(B--C); draw(C--D,green); draw(D--E); draw(E--F); draw(F--A); draw(A--C,red); draw(A--Q); draw(A--E,red); draw(B--E); draw(C--E,red); draw(Q--F); draw(Q--C,green); draw(Q--D,green); draw(circle((0,0),1)); label("\(A\)",A,W); label("\(B\)",B,N); label("\(C\)",C,N); label("\(D\)",D,NE); label("\(E\)",E,SE); label("\(F\)",F,S); label("\(P\)",(0.3,0.8),S); label("\(Q\)",(-0.15,0.4),S); [/asy]

$\angle AEC=\angle ADC$ because $A$,$C$,$D$ and $E$ all lie on the circle, and $\angle ADC=\angle CDQ$. $\angle EAD=\angle ECD$ because $A$,$C$,$D$ and $E$ all lie on the circle. $\angle DAC=\angle ECF$ because $CD=EF$, and $A$,$C$,$D$,$E$ and $F$ all lie on the circle. Then,

$\angle EAC=\angle EAD+\angle DAC=\angle ECD+\angle ECF=\angle DCY$

Therefore, $\triangle AEC$ and $\triangle CDQ$ are similar, so $AC/CE=CQ/QD$.

Lastly, let's show that $\triangle CPQ$ and $\triangle EPD$ are similar.

[asy] pair A,B,C,D,E,F,P,Q; A=(-0.96,0.28); B=(-0.352,0.936); C=(0,1); D=(4/5,3/5); E=(4/5,-3/5); F=(0,-1); P=IntersectionPoint(A--D,C--E); Q=IntersectionPoint(A--D,C--F); draw(A--B); draw(B--C); draw(C--D); draw(D--E,green); draw(E--F); draw(F--A); draw(A--C); draw(A--Q); draw(A--E); draw(B--E); draw(P--E,green); draw(Q--F); draw(C--P,red); draw(Q--P,red); draw(C--Q,red); draw(D--P,green); draw(circle((0,0),1)); label("\(A\)",A,W); label("\(B\)",B,N); label("\(C\)",C,N); label("\(D\)",D,NE); label("\(E\)",E,SE); label("\(F\)",F,S); label("\(P\)",(0.3,0.8),S); label("\(Q\)",(-0.15,0.4),S); [/asy]

Because $CD=EF$ and $C$,$D$,$E$ and $F$ all lie on the circle, $CF$ is parallel to $DE$. So, $\triangle CPQ$ and $\triangle EPD$ are similar, and $CQ/DE=CP/PE$.

Putting it all together, $CP/PE=CQ/DE=AC/CE\cdot QD/DE=(AC/CE)^2$.

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