70 Ceva’s theorem

BP

PC

standard deviation . For example, a factory may

produce light bulbs packaged in large shipping cartons,

100 per carton. Even though the lifespan of individual

light bulbs may vary wildly following no recognizable

distribution of values, the central limit theorem asserts

that the average lifespan of the bulbs per carton is given

by a normal distribution. Since the height of an individ-

ual is the aggregate effect of the growth rate of a large

number of individual cells, the distribution of heights of

men and woman is essentially normal, as is the distribu-

tion of heights of most anything that grows—cats, maple

trees, or carrots, for example.

Another version of the central-limit theorem is use-

ful when trying to ascertain what proportion ppercent

of the entire population possesses a certain property

(such as “Has blood type AB” or “Will vote Republi-

can next November”). As it is impossible to examine

every individual on the globe, or poll every individual

in the nation, one can examine a sample of individuals

and compute the percentage in this sample with the

desired property. The central-limit theorem also asserts:

If many different samples of Nindividuals are

examined, then the distribution of the percent-

ages of those samples possessing a particular

property very closely follows a normal distri-

bution (and the larger the value of N, the bet-

ter is the approximation to a normal curve).

This distribution has mean p, the true percent-

age of the population with this property, and

standard deviation .

Both versions of the central-limit theorem allow statis-

ticians to make inferences and predictions based on sta-

tistical data.

See also G

EORGE

P

ÓLYA

;

STATISTICS

:

INFERENTIAL

.

Ceva’s theorem Let P, Q, and Rbe, respectively,

points on sides BC, CA, and AB of a triangle ABC.

(One is permitted to extend one or more sides of the

triangle.) Then the lines connecting Pto A, Q to B, and

Rto Care

CONCURRENT

if, and only if:

Here BP, for instance, represents the distance between

points Band P, and the ratio is considered positive

if the direction from Bto Pis the same as the direc-

tion from Pto C, and negative if they are in opposite

directions.

This result is due to Italian mathematician Giovanni

Ceva (1647–1734). It is equivalent to the statement that

the operation of finding the

CENTER OF GRAVITY

of point

masses is

ASSOCIATIVE

. For instance, given three masses

at locations A, B, and Cin a plane, one could locate the

center of mass of the entire system by first computing

the center of mass of just two points and then compute

the center of mass of that result with the third point.

Ceva proved that the same final result is produced no

matter which two points are chosen initially.

Ceva’s theorem can be proved mathematically by

making repeated use of M

ENELAUS

’

S THEOREM

.

chain rule (function of a function rule) If y= f(u) is

a function of a quantity u, which in turn is a function

of another quantity x, u = g(x) say, then yitself can be

thought of as a function of xas a

COMPOSITION

of

functions: y= f(g(x)).The chain rule states that the rate

of change of ywith respect to the quantity xis given by

the formula:

This can also be written: (f(g(x)))′= f

′

(g(x))·g

′

(x).For

example, to differentiate y= (x2+ 2)100, we can write

y= u100, where u= x2+ 2 and so:

The chain rule can be proved by making use of the for-

mal definition of a derivative as a

LIMIT

: