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Difference between revisions of "2019 USAJMO Problems/Problem 3"

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(→‎Solution)
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Note that only one point <math>P</math> satisfies the given angle condition. With this in mind, construct <math>P'</math> with the following properties:
 
Note that only one point <math>P</math> satisfies the given angle condition. With this in mind, construct <math>P'</math> with the following properties:
  
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<math>\dot </math>AP' \cdot AB = AD^2<math>
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1) <math>AP' \cdot AB = AD^2</math>
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</math>\dot BP' \cdot AB = CD^2<math>
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2) <math>BP' \cdot AB = CD^2</math>
  
  
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Claim:</math>P = P'<math>
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Claim:<math>P = P'</math>
  
 
Proof:
 
Proof:
  
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The conditions imply the similarities </math>ADP \sim ABD<math> and </math>BCP \sim BAC<math> whence </math>\measuredangle APD = \measuredangle BDA = \measuredangle BCA = \measuredangle CPB<math> as desired. </math>\square<math>
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The conditions imply the similarities <math>ADP \sim ABD</math> and <math>BCP \sim BAC</math> whence <math>\measuredangle APD = \measuredangle BDA = \measuredangle BCA = \measuredangle CPB</math> as desired. <math>\square</math>
  
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Claim: </math>PE<math> is a symmedian in </math>AEB<math>
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Claim: <math>PE</math> is a symmedian in <math>AEB</math>
  
 
Proof:
 
Proof:
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<cmath>\iff \left(\frac{BC}{AD} \right)^2 = \left(\frac{BE}{AE} \right)^2 = \frac{BP}{AP} </cmath>
 
<cmath>\iff \left(\frac{BC}{AD} \right)^2 = \left(\frac{BE}{AE} \right)^2 = \frac{BP}{AP} </cmath>
  
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as desired. </math>\square<math>
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as desired. <math>\square</math>
  
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Since </math>P<math> is the isogonal conjugate of </math>N<math>, </math>\measuredangle PEA = \measuredangle MEC = \measuredangle BEN<math>. However </math>\measuredangle MEC = \measuredangle BEN<math> implies that </math>M<math> is the midpoint of </math>CD<math> from similar triangles, so we are done. </math>\square$
+
Since <math>P</math> is the isogonal conjugate of <math>N</math>, <math>\measuredangle PEA = \measuredangle MEC = \measuredangle BEN</math>. However <math>\measuredangle MEC = \measuredangle BEN</math> implies that <math>M</math> is the midpoint of <math>CD</math> from similar triangles, so we are done. <math>\square</math>
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 12:35, 25 June 2019

Problem

$(*)$ Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD=\angle BPC$. Show that line $PE$ bisects $\overline{CD}$.

Solution

Let $PE \cap DC = M$. Also, let $N$ be the midpoint of $AB$.

Note that only one point $P$ satisfies the given angle condition. With this in mind, construct $P'$ with the following properties:

1) $AP' \cdot AB = AD^2$ 2) $BP' \cdot AB = CD^2$


Claim:$P = P'$

Proof:

The conditions imply the similarities $ADP \sim ABD$ and $BCP \sim BAC$ whence $\measuredangle APD = \measuredangle BDA = \measuredangle BCA = \measuredangle CPB$ as desired. $\square$

Claim: $PE$ is a symmedian in $AEB$

Proof:

We have

\[AP \cdot AB = AD^2 \iff AB^2 \cdot AP = AD^2 \cdot AB\] \[\iff \left( \frac{AB}{AD} \right)^2 = \frac{AB}{AP}\] \[\iff \left( \frac{AB}{AD} \right)^2 - 1 = \frac{AB}{AP} - 1\] \[\iff \frac{AB^2 - AD^2}{AD^2} = \frac{BP}{AP}\] \[\iff \left(\frac{BC}{AD} \right)^2 = \left(\frac{BE}{AE} \right)^2 = \frac{BP}{AP}\]

as desired. $\square$

Since $P$ is the isogonal conjugate of $N$, $\measuredangle PEA = \measuredangle MEC = \measuredangle BEN$. However $\measuredangle MEC = \measuredangle BEN$ implies that $M$ is the midpoint of $CD$ from similar triangles, so we are done. $\square$

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

See also

2019 USAJMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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