MATHEMATICAL NOTES
EDITED BY E. F. BECKENBACIE,
University
of California
Material for this department should be sent directly to E. F. Beckenbach, University of
California,
Los
Angeles
24,
California.
NEW PROOF OF A MINIMUM PROPERTY OF THE REGULAR nGON
L.
F. ToTH, Budapest, Hungary
J. Kurschik gives in his paper
Ober
dem Kreis
ein
und urmgeschriebene
Vielecke*
among others
a
complete and entirely elementary geometrical proof
of the well known fact according to which the regular ngon Pn has
a
minimal
area among all ngons
P
circumscribed about a circle c. In this proof
P.
is
carried,
after
a
dismemberment
and
a suitable reassembly,
in n
I
steps
into
Pn
so
that
the
area increases at
every
step.
In
this note we give an extremely simple proof,t which appears
to
be new,
showing immediately that if Pn is not regular, then P,
,,,
where
the area
is
denoted by the same symbol as the domain.
Consider the circle C circumscribed about P. We
show
that already
for
the
part Pn C of
P.
lying in C we have
PnC
>
Pn.
We
have Pn
C=Cns+(S1S2+S2s3+
**
+SnSI),
where
we
denote
by
S, S2,
S* *
,
sn
the circular sections of C cut off by the consecutive sides of
Pn,
and
by s the circular section of C cut off by a tangent to c. Hence
Pn*C
_ C  nS.
Then
P7 >!
Pn
C
>
C

ns
=Pn.
Equality holds
in
P.
r!
P.C
resp. in
PnC
?
Pn
only if no vertex
of
Pn
lies
in
the
outside
resp.
in
the
inside
of
C;
this
completes
the
proof.
BINOMIAL
COEFFICIENTS MODULO A PRIME
N. J. FINE,
University of
Pennsylvania
The
following
theorem,
although
given by
Lucas
in
his
Theorie
des
Nombres
(pp.
417420),
does not
appear
to be as
widely
known as
it
deserves
to
be:
THEOREM 1.
Let p be
a prime,
and let
M
=
MO
+
MIP
+
M2p2
+
*
.
+
Mkpk
(O
<
Mr
<
P),
No
+ Np + N2p2+
**
+Nkpk
(O
5
Nr
<
P).
*
Mathematische Annalen 30 (1887), pp. 578581.
t
As
P. Sz6sz
remarked [Bemerkung zu einer
Arbeit von K.
Ktirschik,
Matematikai
es
Fizikai Lapok
XLIV (1937), p. 167, note 3]
Kurschik's
proof is independent of the axiom
of
parallels.
This
advantage
is
preserved in the present
proof.
589
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