median of a triangle 335

Medians are concurrent

g

g

g

g

g

The mean-value theorem itself can be established

as a consequence of R

OLLE

’

S THEOREM

. One does this

by writing down the equation of the line that connects

the two endpoints (a,f(a)) and (b,f (b)) of the function f.

It is given by:

(Put in x= aand x= bto see that this is correct.) Now

consider the function:

h(k) = f(x) – g(x)

It is a differentiable function with h(a) = h(b) = 0, and

so by Rolle’s theorem, there is at least one value c

between aand bfor which h′(c) = f′(c) – g′(c) = 0. This

yields the equation asserted in the statement of the

mean-value theorem.

The mean-value theorem can be thought of as a

statement about the nature of a differentiable curve

intersecting a straight line. French mathematician

A

UGUSTIN

-L

OUIS

C

AUCHY

(1789–1857) later general-

ized the theorem to one about any two differentiable

curves intersecting at two points. The result is known

as the extended mean-value theorem or Cauchy’s mean-

value theorem.

If two functions fand ghave the same values

at x= aand x= b, are continuous in the closed

interval [a,b], differentiable in the open inter-

val (a,b),and further if g(a) ≠g(b) and g′(x) is

never zero in (a,b),then there is at least one

value cbetween aand bfor which:

The theorem is proved in a similar way by making use

of the support function:

See also

DERIVATIVE

;

INCREASING

/

DECREASING

.

median See

STATISTICS

:

DESCRIPTIVE

.

median of a triangle A line segment connecting the

MIDPOINT

of one side of a triangle to the vertex opposite

to that side is called a median of the triangle. Any trian-

gle has three medians. It is considered a fundamental

result that the three medians of a triangle always meet at

a common point (called the centroid of the triangle and

usually denoted G.) To see this, consider a triangle with

vertices A, B, and Cand midpoints as shown.

First note:

Any line connecting the midpoints of two

sides of a triangle is parallel to the third side

of the triangle.

In the diagram above, by the SAS principle, triangles

BAC and MCAMBare similar, with a scale factor of

two, since they share a common angle at Aand the two

sides of each triangle match in a 2-to-1 ratio. Conse-

quently the angles labeled xare equal, yielding two

equal alternate angles, from which it follows from the

converse of the

PARALLEL POSTULATE

that the lines

MCMBand BC are parallel.

We now have that angles MBMCC and MCCB are

equal, as are angles MCMBB and MBBC. Consequently,

by the AAA principle, the two shaded triangles are sim-

ilar, again in a 2-to-1 ratio. In particular the line seg-

ments BG and GMBare in this ratio, as are the line

segments CG and GMC. This establishes:

The point of intersection of any two medians

of a triangle lies two-thirds of the way along

each median.

Consequently, the median AMAwill also intercept

median BMBtwo-thirds the distance along the length

of BMB, namely, at the same point G. Thus all three

medians are indeed

CONCURRENT

at G.