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To That End, Let Us Call A Point P A Point of Equal Potency, If For

1. The document discusses a property of circles and spheres called equal potency, where the product of distances from a point to any two points on a chord passing through it is constant. 2. It shows that having one point of equal potency is not enough to characterize a circle, using an example of an equilateral triangle. 3. It proves that if a simple closed curve has one chord of equal potency, then the curve must be a circle. 4. It extends this to surfaces in three dimensions, proving that if a closed convex surface has one plane of equal potency, then the surface must be a sphere.

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62 views2 pages

To That End, Let Us Call A Point P A Point of Equal Potency, If For

1. The document discusses a property of circles and spheres called equal potency, where the product of distances from a point to any two points on a chord passing through it is constant. 2. It shows that having one point of equal potency is not enough to characterize a circle, using an example of an equilateral triangle. 3. It proves that if a simple closed curve has one chord of equal potency, then the curve must be a circle. 4. It extends this to surfaces in three dimensions, proving that if a closed convex surface has one plane of equal potency, then the surface must be a sphere.

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Va Lentín
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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On a Characteristic Property of the Circle

and the Sphere,

by

KITIZI YANAGmABA in Sendai.

1. It is well known that if a variable chord MN of a circle (or


a sphere), which passes through a given point P, the potency of P
with respect to the circle (or the sphere) is constant irrespective of the
direction of MN.
In the following lines, it will be discussed whether this property
be characteristic or not.
To that end, let us call a point P a point of equal potency,if for
every chord MN passing through P, PM.PN is constant, and also let
MN be called a chord of equal potency, if every point of it is of equal
potency.
2. The existence of only one point of equal potency is not sufficient
to characterize the circle.
This can at once be seen from the following example.
Let the three sides BO, CA, AB of an equilateral triangle ABO
be divided in three equal parts by A1,A2; B1,B2;
C1,C2 respectively. Then B1C2, CIA, and AjB,
meet in the point 0, centre of the triangle
ABC Nowit willeasilybe seenthat the circles
OB1C2, 002A1 and OA,B, are respectively the in-
circles of the equilateral triangles AB102, BOA .,
0A1B2, and also that , the inferior arcs B,01, 02A1
and ABj are respectively the inverses of the segments AIAB1B2,
01C, with respect to the circle K having centre 0, and radius VTAB/s.
Therefore 0 is a point of equal potency of the simply closed
convex curve A1A,B1B,0102
3. If a simply closed convex curve r has at least a
chord of equal potency, then r must be a circle.
Let MN be a chord of equal potency of r, and
AB be any one, chord divided internally by MN in a
point P. Then we get
ON A CHARACTERISTICPROPERTY OF THE CIRCLE AND THE SPHERE. 143

MP.PN=AP.PB.
Hence A,B,M and N lie on one and the same circle K.
Next let a be any point of the arc MBN, and let the point of
intersection of MN and AO be denoted by Q. Then since we have
AQ.QO=MQ.QN,
0 must be a point on the circle AMN, i.e. K.
By just the same way, it can be proved that every point of the
are MAN is situated on the circle BMN, i.e.K. Therefore r must
be coincident with the circle K.
4. Next let us call a plane section of a simply closed convex
surface F a plane of equal potency, if every point of it is of equal
potency with respect to F. Then we can prove the following
Theorem: If, in a simply closed convex swfaces F, there exist a
plane of equal potency, then F must be a sphere.
Let n be a plane of equal potency of F, which cuts F along a
simply closed convex curve 1. Then since any
chord of 1 is of equal potency, r must be a circle.
Take any point A on 1, and draw a supporting
line (Stiite) t through A, cutting the plane n in
A. Draw any plane n, passing through t and
cutting r along a simply closed convex curve r'.
If the other point of intersection of r and F' is
denoted by P, AP is evidently a chord of equal
potency of F', and therefore r, must also be a circle, touching tatA.
Then by turning the plane ir' continuously about t, it can be conclud-
ed that F is a sphere, by means of the lemma 2 in my note "A
theorem on surface" in this Journal, Vol.8, p. 42.

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