MathDB

We define the distance between two circles

\omega ,\omega '

by the length of the common external tangent of the circles and show it by

d(\omega , \omega ')

. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius

0

and the distance between two cirlces can be zero. (a) Centroid.

n

circles

\omega_1,\dots, \omega_n

are fixed on the plane. Prove that there exists a unique circle

\overline \omega

such that for each circle

\omega

on the plane the square of distance between

\omega

and

\overline \omega

minus the sum of squares of distances of

\omega

from each of the

\omega_i

1\leq i \leq n

is constant, in other words:

d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant

(b) Perpendicular Bisector. Suppose that the circle

\omega

has the same distance from

\omega_1,\omega_2

. Consider

\omega_3

a circle tangent to both of the common external tangents of

\omega_1,\omega_2

. Prove that the distance of

\omega

from centroid of

\omega_1 , \omega_2

is not more than the distance of

\omega

and

\omega_3

. (If the distances are all defined) (c) Circumcentre. Let

C

be the set of all circles that each of them has the same distance from fixed circles

\omega_1,\omega_2,\omega_3

. Prove that there exists a point on the plane which is the external homothety center of each two elements of

C

. (d) Regular Tetrahedron. Does there exist 4 circles on the plane which the distance between each two of them equals to

1

?
Time allowed for this problem was 150 minutes.

Problem Statement