Question 25
The **distance** between two circles $\omega$ and $\omega'$ is defined as the length of their common external tangent and is represented as $d(\omega, \omega')$. If two circles don't have a common external tangent, the distance between them is not defined. Also, note that a point is a circle with zero radius and that the distance between two circles can be zero.
a) **Centroid.** Circles $\omega_1, \omega_2, ..., \omega_n$ are given in the plane where $n$ is a natural number. Prove that a unique circle $\bar{\omega}$ exists such that for an arbitrary circle $\omega$ in that plane, the difference between the square of the distance between $\omega$ and $\bar{\omega}$ and the average of the squares of the distances between $\omega$ and $\omega_i$ ($1 \le i \le n$) is constant (for those circles $\omega$ that all these distances are defined). That is,
$$
\forall \omega : d(\omega, \bar{\omega})^2 - \frac{1}{n} \sum_{i=1}^{n} d(\omega, \omega_i)^2 = \text{constant.}
$$
$\bar{\omega}$ is called the **centroid** of these circles, for it is similar to the centroid of $n$ points in the plane.
b) **Perpendicular bisector.** Suppose that circle $\omega$ is equidistant from circles $\omega_1$ and $\omega_2$. Let $\omega_3$ be an arbitrary circle whose center is on the centerline of circles $\omega_1$ and $\omega_2$ and is tangent to the common external tangent of circles $\omega_1$ and $\omega_2$. Prove that "the distance between $\omega$ and the centroid of circles $\omega_1$ and $\omega_2$" is not more than "the distance between $\omega$ and $\omega_3$" (for the case that all these distances are defined).
c) **Circumcenter.** Let $C$ be the set of all circles in the plane that each one of them is equidistant from three fixed circles $\omega_1, \omega_2$ and $\omega_3$. Prove that a fixed point exists in the plane that is the direct homothetic center of each two circles in $C$.
d) **Regular tetrahedron.** Do there exist four circles in the plane for which the distance between any two of them is unity?
150 minutes (→ p.34)
a) **Centroid.** Circles $\omega_1, \omega_2, ..., \omega_n$ are given in the plane where $n$ is a natural number. Prove that a unique circle $\bar{\omega}$ exists such that for an arbitrary circle $\omega$ in that plane, the difference between the square of the distance between $\omega$ and $\bar{\omega}$ and the average of the squares of the distances between $\omega$ and $\omega_i$ ($1 \le i \le n$) is constant (for those circles $\omega$ that all these distances are defined). That is,
$$
\forall \omega : d(\omega, \bar{\omega})^2 - \frac{1}{n} \sum_{i=1}^{n} d(\omega, \omega_i)^2 = \text{constant.}
$$
$\bar{\omega}$ is called the **centroid** of these circles, for it is similar to the centroid of $n$ points in the plane.
b) **Perpendicular bisector.** Suppose that circle $\omega$ is equidistant from circles $\omega_1$ and $\omega_2$. Let $\omega_3$ be an arbitrary circle whose center is on the centerline of circles $\omega_1$ and $\omega_2$ and is tangent to the common external tangent of circles $\omega_1$ and $\omega_2$. Prove that "the distance between $\omega$ and the centroid of circles $\omega_1$ and $\omega_2$" is not more than "the distance between $\omega$ and $\omega_3$" (for the case that all these distances are defined).
c) **Circumcenter.** Let $C$ be the set of all circles in the plane that each one of them is equidistant from three fixed circles $\omega_1, \omega_2$ and $\omega_3$. Prove that a fixed point exists in the plane that is the direct homothetic center of each two circles in $C$.
d) **Regular tetrahedron.** Do there exist four circles in the plane for which the distance between any two of them is unity?
150 minutes (→ p.34)