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Difference between revisions of "1992 USAMO Problems/Problem 4"

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==Solution==
 
==Solution==
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===Solution 1===
 
Consider the plane through <math>A,A',B,B'</math>.  This plane, of course, also contains <math>P</math>. We can easily find the <math>\triangle APB</math> is isosceles because the base angles are equal. Thus, <math>AP=BP</math>. Similarly, <math>A'P=B'P</math>. Thus, <math>AA'=BB'</math>. By symmetry, <math>BB'=CC'</math> and <math>CC'=AA'</math>, and hence <math>AA'=BB'=CC'</math> as desired.  
 
Consider the plane through <math>A,A',B,B'</math>.  This plane, of course, also contains <math>P</math>. We can easily find the <math>\triangle APB</math> is isosceles because the base angles are equal. Thus, <math>AP=BP</math>. Similarly, <math>A'P=B'P</math>. Thus, <math>AA'=BB'</math>. By symmetry, <math>BB'=CC'</math> and <math>CC'=AA'</math>, and hence <math>AA'=BB'=CC'</math> as desired.  
  
 
<math>\mathbb{QED.}</math>
 
<math>\mathbb{QED.}</math>
  
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=== Add-on===
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==== Add-on====
 
By another person ^v^
 
By another person ^v^
  
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Since any cross section of sphere is a circle. It implies that <math>A</math>, <math>A'</math>, <math>B</math>, <math>B'</math> be on the same circle (<math>\omega_1</math>), <math>A</math>, <math>B</math>, <math>P</math> be on the same circle (<math>\omega_2</math>), and <math>A'</math>, <math>B'</math>, <math>P</math> be on the same circle (<math>\omega_3</math>).
 
Since any cross section of sphere is a circle. It implies that <math>A</math>, <math>A'</math>, <math>B</math>, <math>B'</math> be on the same circle (<math>\omega_1</math>), <math>A</math>, <math>B</math>, <math>P</math> be on the same circle (<math>\omega_2</math>), and <math>A'</math>, <math>B'</math>, <math>P</math> be on the same circle (<math>\omega_3</math>).
  
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<math>m\angle APB= m\angleA'PB'</math> because they are vertical angles. By power of point, <math>(AP)(A'P)=(BP)(B'P)\rightarrow\frac{AP}{B'P}=\frac{BP}{A'P}</math>
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<math>m\angle APB= m\angle A'PB'</math> because they are vertical angles. By power of point, <math>(AP)(A'P)=(BP)(B'P)\rightarrow\frac{AP}{B'P}=\frac{BP}{A'P}</math>
  
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By the SAS triangle simlarity theory, <math>\triangle APB \sim\triangle B'PA'</math>. That implies that <math>\angle ABP\cong\angleB'PA'</math>.
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By the SAS triangle simlarity theory, <math>\triangle APB \sim\triangle B'PA'</math>. That implies that <math>\angle ABP\cong\angle B'PA'</math>.
  
 
<br/>
 
<br/>
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<math>l</math> must be the common tangent of <math>\omega_2</math> and <math>\omega_3</math>.
 
<math>l</math> must be the common tangent of <math>\omega_2</math> and <math>\omega_3</math>.
  
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The acute angles form by <math>l</math> and <math>\overbar{AA'}</math> are congruent to each other (vertical angles) and by the tangent-chord theorem, the central angle of chord <math>\overbar{AP}</math> and <math>\overbar{A'P}</math> are equal.
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The acute angles form by <math>l</math> and <math>\overline{AA'}</math> are congruent to each other (vertical angles) and by the tangent-chord theorem, the central angle of chord <math>\overline{AP}</math> and <math>\overline{A'P}</math> are equal.
  
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Similarly the central angle of chord <math>\overbar{BP}</math> and <math>\overbar{B'P}</math> are equal.
+
Similarly the central angle of chord <math>\overline{BP}</math> and <math>\overline{B'P}</math> are equal.
  
 
The length of any chord with central angle <math>2\theta</math> and radius <math>r</math> is <math>2r\sin\left({\theta}\right)</math>, which can easily been seen if we drop the perpendicular from the center to the chord.
 
The length of any chord with central angle <math>2\theta</math> and radius <math>r</math> is <math>2r\sin\left({\theta}\right)</math>, which can easily been seen if we drop the perpendicular from the center to the chord.
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Thus, <math>\frac{AP}{A'P}=\frac{BP}{B'P}</math>.
 
Thus, <math>\frac{AP}{A'P}=\frac{BP}{B'P}</math>.
  
−
By the SAS triangle simlarity theory, <math>\triangle APB \sim\triangle A'PB'</math>. That implies that <math>\angle ABP\cong\angleB'PA'</math>.
+
By the SAS triangle simlarity theory, <math>\triangle APB \sim\triangle A'PB'</math>. That implies that <math>\angle ABP\cong\angle B'PA'</math>.
  
 
<br/>
 
<br/>
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That implies that <math>\angle ABP\cong\angleA'PB'\cong\angleB'PA'</math>. Thus, <math>\triangle A'PB'</math> is an isosceles triangle and since <math>\triangle APB \sim\triangle A'PB'</math>,<math>\triangle APB</math> is an isosceles triangle too.
+
That implies that <math>\angle ABP\cong\angle A'PB'\cong\angle B'PA'</math>. Thus, <math>\triangle A'PB'</math> is an isosceles triangle and since <math>\triangle APB \sim\triangle A'PB'</math>, then<math>\triangle APB</math> must be an isosceles triangle too.
  
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== Resources ==
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===Solution 2===
  
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{{USAMO box|year=1992|num-b=3|num-a=5}}
 
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks]
 
  
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Call the large sphere <math> O_1</math>, the one containing <math>A</math> <math> O_2</math>, and the one containing <math> A'</math> <math>O_3</math>.  The centers are <math> c_1</math>, <math> c_2,</math> and <math> c_3</math>.
 +
 +
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Since two spheres always intersect in a circle , we know that A,B, and C must lie on a circle (<math> w_1</math>)completely contained in <math> O_1</math> and <math> O_2</math>
 +
 +
 +
Similarly, A', B', and C' must lie on a circle (<math>w_2</math>) completely contained in <math> O_1</math> and <math> O_3</math>. 
 +
 +
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So, we know that 3 lines connecting a point on <math> w_1</math> and P hit a point on <math> w_2</math>.  This implies that <math> w_1</math> projects through P to <math> w_2</math>, which in turn means that <math> w_1</math> is in a plane parallel to that of <math> w_2</math>.  Then, since <math> w_1</math> and <math> w_2</math> lie on the same sphere, we know that they must have the same central axis, which also must contain P (since the center projects through P to the other center). 
 +
 +
 +
So, all line from a point on <math> w_1</math> to P are of the same length, as are all lines from a point on <math> w_2</math> to P.  Since AA', BB', and CC' are all composed of one of each type of line, they must all be equal.
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 +
== See Also ==
  
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[[Category:Olympiad Algebra Problems]]
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{{USAMO box|year=1992|num-b=3|num-a=5}}
 +
{{MAA Notice}}
 +
[[Category:Olympiad Geometry Problems]]
 +
[[Category:3D Geometry Problems]]

Latest revision as of 17:37, 25 June 2021

Problem

Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$.

Solution

Solution 1

Consider the plane through $A,A',B,B'$. This plane, of course, also contains $P$. We can easily find the $\triangle APB$ is isosceles because the base angles are equal. Thus, $AP=BP$. Similarly, $A'P=B'P$. Thus, $AA'=BB'$. By symmetry, $BB'=CC'$ and $CC'=AA'$, and hence $AA'=BB'=CC'$ as desired.

$\mathbb{QED.}$

Add-on

By another person ^v^

The person that came up with the solution did not prove that $\triangle APB$ is isosceles nor the base angles are congruent. I will add on to the solution.

There is a common tangent plane that pass through $P$ for the $2$ spheres that are tangent to each other.


Since any cross section of sphere is a circle. It implies that $A$, $A'$, $B$, $B'$ be on the same circle ($\omega_1$), $A$, $B$, $P$ be on the same circle ($\omega_2$), and $A'$, $B'$, $P$ be on the same circle ($\omega_3$).

$m\angle APB= m\angle A'PB'$ because they are vertical angles. By power of point, $(AP)(A'P)=(BP)(B'P)\rightarrow\frac{AP}{B'P}=\frac{BP}{A'P}$

By the SAS triangle simlarity theory, $\triangle APB \sim\triangle B'PA'$. That implies that $\angle ABP\cong\angle B'PA'$.


Let's call the interception of the common tangent plane and the plane containing $A$, $A'$, $B$, $B'$, $P$, line $l$.

$l$ must be the common tangent of $\omega_2$ and $\omega_3$.

The acute angles form by $l$ and $\overline{AA'}$ are congruent to each other (vertical angles) and by the tangent-chord theorem, the central angle of chord $\overline{AP}$ and $\overline{A'P}$ are equal.

Similarly the central angle of chord $\overline{BP}$ and $\overline{B'P}$ are equal.

The length of any chord with central angle $2\theta$ and radius $r$ is $2r\sin\left({\theta}\right)$, which can easily been seen if we drop the perpendicular from the center to the chord.

Thus, $\frac{AP}{A'P}=\frac{BP}{B'P}$.

By the SAS triangle simlarity theory, $\triangle APB \sim\triangle A'PB'$. That implies that $\angle ABP\cong\angle B'PA'$.


That implies that $\angle ABP\cong\angle A'PB'\cong\angle B'PA'$. Thus, $\triangle A'PB'$ is an isosceles triangle and since $\triangle APB \sim\triangle A'PB'$, then$\triangle APB$ must be an isosceles triangle too.

Solution 2

Call the large sphere $O_1$, the one containing $A$ $O_2$, and the one containing $A'$ $O_3$. The centers are $c_1$, $c_2,$ and $c_3$.


Since two spheres always intersect in a circle , we know that A,B, and C must lie on a circle ($w_1$)completely contained in $O_1$ and $O_2$


Similarly, A', B', and C' must lie on a circle ($w_2$) completely contained in $O_1$ and $O_3$.


So, we know that 3 lines connecting a point on $w_1$ and P hit a point on $w_2$. This implies that $w_1$ projects through P to $w_2$, which in turn means that $w_1$ is in a plane parallel to that of $w_2$. Then, since $w_1$ and $w_2$ lie on the same sphere, we know that they must have the same central axis, which also must contain P (since the center projects through P to the other center).


So, all line from a point on $w_1$ to P are of the same length, as are all lines from a point on $w_2$ to P. Since AA', BB', and CC' are all composed of one of each type of line, they must all be equal.

See Also

1992 USAMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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