In three-dimensional space, the set of all points
equidistant from a single point Ois a sphere with Oas
its center. Given two points Aand B, the set of all
points equidistant from them is a plane that passes
through the midpoint of the line segment AB and is
perpendicular to it. Given three points A, B, and C, not
all in a straight line, the set of all points Pequidistant
from all three is a straight line perpendicular to the
plane formed by the three points and passing through
the circumcenter of the triangle ABC. There need not
be a point equidistant from four given points in three-
dimensional space.
The distance of a point from a line is the length of
a line segment from the point meeting that line at right
angles. Again, using Pythagoras’s theorem and similar
triangles, one can show that the set of all points
equidistant from two intersecting lines in a plane is a
pair of perpendicular lines that each pass through the
point of intersection of the two lines, and each bisects
an angle formed by the lines. Furthermore, arguing as
above, one can prove:
In any triangle, the three lines that bisect the
interior angles of the triangle meet at a com-
mon point.
(Consider first the point of intersection of just two
angle bisectors. This point must, in fact, be equidistant
from all three sides of the triangle, and so lies on the
third angle bisector.)
See also E
ULER LINE
;
INCIRCLE
.
equilateral A
POLYGON
is said to be equilateral if all
of its sides have the same length. For example, a square
is equilateral, as is a triangle with each interior angle
equal to 60°. A polygon is called “regular” if it is both
equilateral and
EQUIANGULAR
.
A point (x,y) in the Cartesian plane is said to be a
“lattice point” if both xand yare integers, and a poly-
gon drawn in the plane is said to be a “lattice polygon”
if its vertices lie at lattice points. Mathematicians have
proved that it is impossible to draw an equilateral lat-
tice polygon with an odd number of sides, although
equilateral lattice polygons with any even number of
sides do exist. The square and the octagon are the only
two regular lattice polygons.
See also C
ARTESIAN COORDINATES
.
equivalence relation See
PAIRWISE DISJOINT
.
Eratosthenes of Cyrene (ca. 275–195
B
.
C
.
E
.) Greek
Geometry, Number theory, Astronomy, Geographer
Born in Cyrene, in North Africa, (the exact birth date
is not known), Eratosthenes is remembered as the first
person to calculate the circumference of the Earth. (See
E
ARTH
.) Using the known distance between two partic-
ular cities, the lengths of shadows cast by the noonday
sun at those cities, and simple geometric reasoning,
Eratosthenes determined the circumference of the Earth
to be 250,000 “stadia.” Unfortunately, the exact length
of a “stade” is not known today, and so it is not possi-
ble to be certain of the accuracy of this result. If we
take, as many historians suggest, that the likely length
of this unit is 515.6 ft (157.2 m), then Eratosthenes’
calculation is extraordinarily accurate.
Eratosthenes traveled to Athens in his youth and
spent many years studying there. Around 240
B
.
C
.
E
.he
was appointed librarian of the greatest library of the
ancient world, the Library of Alexandria. Early in his
scholarly career, Eratosthenes wrote the expository
piece Platonicus as an attempt to explain the mathe-
matics on which P
LATO
based his philosophy. Although
this work is lost today, scholars of later times referred
to it frequently and described it as an invaluable source
detailing the mathematics of geometry and arithmetic,
as well as the mathematics of music. In this work,
Eratosthenes also described the problem of
DUPLICAT
-
ING THE CUBE
and provided a solution to it making use
of a mechanical device he invented.
Eratosthenes also worked on the theory of
PRIME
numbers and discovered a famous “sieve” technique
for finding primes. This method is still used today and
is named in his honor.
Along with measuring the circumference of the
Earth, Eratosthenes also devised ingenious techniques
for determining the distance of the Earth from the Sun
(which he measured as 804 million stadia), the distance
between the Earth and the Moon (780,000 stadia), and
the tilt of the Earth’s axis with respect to the plane in
which the Earth circles the Sun (which he measured as
11/83 of 180°, that is, 23°51′15′′). Eratosthenes also
accurately mapped a significant portion of the Nile
River and correctly identified the occurrence of heavy
rains near its source as the reason for its erratic flood-
ing near its mouth. He compiled an astronomical cata-
166 equilateral