%\date 05.08.06

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\begin{document}
\title{Theorem on altitudes and the Jacobi identity}
\author{A. Zaslavskiy and M. Skopenkov}
\date{}
\maketitle

In this subject we shall show how to use the Arnold idea to obtain
theorems not only on points and lines but also on circles. Let us
give one example:

\smallskip
\noindent{\bf Theorem on bisectors of a curved triangle.} Let $a$,
$b$ and $c$ be three pairwise intersecting circles in the plane.
Draw a circle $c'$ passing through both intersection points of the
circles $a$ and $b$ such that the angle between $c'$ and $a$ is
equal to the angle between $c'$ and $b$. (There are two circles
satisfying these conditions, choose the one having common points
with the interior of the intersection of the areas bounded by $a$
and $b$.) Define the circles $a'$ and $b'$ analogously. Then the
three circles $a'$, $b'$ and $c'$ have a common point.
\smallskip

\flushpar{\bf 27.} Prove the theorem on bisectors of a curved
triangle.

\flushpar{\bf 28*.} Fix a circle $I$ in the plane. By a {\it
common perpendicular} to a pair of circles $a$ and $b$ we mean a
circle perpendicular to all the three circles $a$, $b$ and $I$.
Prove that Theorem on altitudes of a 'triside' remains true, if
one replaces the word 'line' (in space) by 'circle' (in the plane)
everywhere in the statement.

\smallskip

Let us start again with spherical geometry, in which the Arnold
idea is revealed most clearly. The problems of this subject can be
solved independently from the others.

\smallskip
{\bf Subject Three. Summing of circles.}
\smallskip

By a {\it circle} we mean the section of the sphere by an
arbitrary plane, not necessarily passing trough the center of the
sphere. To each circle assign a vector perpendicular to this
plane, looking toward the plane and having length $1/h$, where $h$
is the distance between the plane and the center of the sphere.

In what follows uppercase letters always denote circles. The
vector corresponding to a circle is denoted by the same letter as
the circle, and, to distinguish them, with the symbol of the
vector. For a vector $\vec A$ denote by $d_A$ the length of a
tangent to our sphere, dropped from the end of the vector $\vec
A$.

\smallskip

 \flushpar{\bf 29.} What is the geometric sense of the condition
$\vec A+\vec B+\vec C=0$?

\flushpar{\bf 30.} Prove the formula $(\vec A,\vec A)=1+d_A^2$.

\flushpar{\bf 31.} What is the geometric sence of the condition
$(\vec A,\vec B)=1$?

\flushpar{\bf 32.} To what circle does the vector
$$
\frac{[\vec A,\vec B]+[\vec B,\vec C]+[\vec C,\vec A]}{(\vec
A,\vec B,\vec C)}
$$
correspond?

\flushpar{\bf 33.} What is the geometric sense of the condition
$\vec A-\vec B+\vec C-\vec D=0$?

\flushpar{\bf 34.} To what circle does the vector
$$
\frac{d_B}{d_B-d_A}\vec A-\frac{d_A}{d_B-d_A}\vec B
$$
correspond?

\flushpar{\bf 35.} Use these facts and algebraic identities to
obtain more geometric theorems!

\smallskip
{\bf Appendix. Presentation of circles by points in space.}
\smallskip

Take a circle in the plane be given by the equation
$x^2+y^2+ax+by+c=0$. To this circle we assign the point in space
having the coordinates $(a,b,c)$.

\noindent{\bf 36.} For which numbers $a$, $b$ and $c$ there is a
circle corresponding to the point $(a,b,c)$?

\noindent{\bf 37.} Consider a circle corresponding to the point
$P$ in space. What is the set of all the points corresponding to
the circles orthogonal to our one?

\noindent{\bf 38.} What pairs of points correspond to a pair of

a)intersecting;

b)non-intersecting;

c)tangent circles?

\noindent{\bf 39.} Prove the following theorem (V. Yu. Protasov).

\noindent Take three circles such that the first is inside the
second and the second is inside the third. Consider chains of
circles tangent to both the first and the third one and such that
one of the intersection points of any two consequent circles in
the chain belong to the second given circle. If this chain is
closed for some starting circle, then it is closed for any choice
of starting circle.

\noindent{\bf 40.} Take a sphere, a plane tangent to the sphere
and four points $A$, $B$, $C$ and $D$ in the plane. Let $D'$ be
the intersection point of three planes passing through the lines
$AB$, $BC$ and $CA$ respectively and tangent to the sphere. Define
the points $A'$, $B'$ and $C'$ analogously. Prove that the points
$A'$, $B'$, $C'$ and $D'$ lie in one plane tangent to the sphere.

\end{document}
