16 Apollonius’s circle

The first volume of The Conics simply reviews ele-

mentary material about the topic and chiefly presents

results already known to Euclid. Volumes two and

three present original results regarding the

ASYMPTOTE

s

to hyperbolas and the construction of

TANGENT

lines to

conics. While Euclid demonstrated a means, for

instance, of constructing a circle passing through any

three given points, Apollonius demonstrated techniques

for constructing circles tangent to any three lines, or to

any three circles, or to any three objects be they a com-

bination of points, lines, or circles. Volumes four, five,

six, and seven of his famous work are highly innovative

and contain original results exploring issues of curva-

ture, the construction of normal lines, and the con-

struction of companion curves to conics. Apollonius

also applied the theory of conics to solve practical

problems. He invented, for instance, a highly accurate

sundial, called a hemicyclium, with hour lines drawn

on the surface of a conic section.

Apollonius also played a fundamental role in the

development of Greek mathematical astronomy. He

proposed a complete mathematical analysis of epicyclic

motion (that is, the compound motion of circles rolling

along circles) as a means to help explain the observed

retrograde motion of the planets across the skies that

had confused scholars of his time.

Apollonius’s work was extraordinarily influen-

tial, and his text on the conics was deemed a stan-

dard reference piece for European scholars of the

Renaissance. J

OHANNES

K

EPLER

, R

ENÉ

D

ESCARTES

,

and S

IR

I

SSAC

N

EWTON

each made reference to The

Conics in their studies.

See also

CIRCUMCIRCLE

;

CYCLOID

.

Apollonius’s circle Let Aand Bbe two points of the

plane and let kbe a constant. Then the set of all points

Pwhose distance from A is k times its distance from B

is a

CIRCLE

. Any circle obtained this way is referred to

as one of Apollonius’s circles. Note that when k= 1 the

circle is “degenerate,” that is, the set of all points

EQUIDISTANT

from Aand Bis a straight line. When k

becomes large, the Apollonius’s circle approaches a cir-

cle of radius 1.

To see that the locus of points described this way

is indeed a circle, set Ato be the origin (0,0), Bto be

the point (k+ 1, 0) on the x-axis, and Pto be a gen-

eral point with coordinates (x,y).The

DISTANCE FOR

-

MULA

then gives an equation of the form

. This is equivalent to

, which is indeed the equa-

tion of a circle, one of radius . A

POLLONIUS OF

P

ERGA

used purely geometric techniques, however, to

establish his claim.

Apollonius’s theorem If a, b, and care the side-

lengths of a triangle and a median of length mdivides

the third side into two equal lengths c/2 and c/2, then

the following relation holds:

This result is known as Apollonius’s theorem. It can be

proved using two applications of the

LAW OF COSINES

as follows:

Let Bbe the

ANGLE

between the sides of length

aand c. Then m2= a2+ (c/2)2–ac cos(B) and b2

= a2+ c2– 2ac cos(B). Solving for ac cos(B) in

the first equation and substituting into the sec-

ond yields the result.

See also

MEDIAN OF A TRIANGLE

.

apothem (short radius) Any line segment from the

center of a regular

POLYGON

to the midpoint of any of

its sides is called an apothem. If the regular polygon

has nsides, each one unit in length, then an exercise in

TRIGONOMETRY

shows that each apothem of the figure

has length .

An analog of

PI

(π) for a regular polygon is the

RATIO

of its

PERIMETER

to twice the length of its

apothem. For a regular n-sided polygon, this ratio has

value ntan(180/n). The

SQUEEZE RULE

shows that this

quantity approaches the value πas nbecomes large.

See also

LONG RADIUS

.