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

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

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. $\mathbb{QED.}$

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 ( $\omega_3$).

$m\angle APB= m\angleA'PB'$ (Error compiling LaTeX. ! Undefined control sequence.) 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 $\angle ABP\cong\angleB'PA'$ (Error compiling LaTeX. ! Undefined control sequence.).

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 $\overbar{AA'}$ (Error compiling LaTeX. ! Undefined control sequence.) are congruent to each other (vertical angles) and by the tangent-chord theorem, the central angle of chord $\overbar{AP}$ (Error compiling LaTeX. ! Undefined control sequence.) and $\overbar{A'P}$ (Error compiling LaTeX. ! Undefined control sequence.) are equal.

Similarly the central angle of chord $\overbar{BP}$ (Error compiling LaTeX. ! Undefined control sequence.) and $\overbar{B'P}$ (Error compiling LaTeX. ! Undefined control sequence.) 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 $\angle ABP\cong\angleB'PA'$ (Error compiling LaTeX. ! Undefined control sequence.).

That implies that $\angle ABP\cong\angleA'PB'\cong\angleB'PA'$ (Error compiling LaTeX. ! Undefined control sequence.). Thus, $\triangle A'PB'$ is an isosceles triangle and since $\triangle APB \sim\triangle A'PB'$, $\triangle APB$ is an isosceles triangle too.

Resources

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