12 altitude

Proving altitudes are concurrent

In 1705, G

OTTFRIED

W

ILHELM

L

EIBNIZ

noticed that

many convergent alternating series, like the Gregory

series, have terms aithat decrease and approach zero:

i. a1≥a2≥a3≥…

ii. an→0

He managed to prove that any alternating series satis-

fying these two conditions does indeed converge, and

today this result is called the “alternating series test.”

(One can see that the test is valid if one physically

paces smaller and smaller steps back and forth: a1feet

forward, a2feet backward, a3feet forward, and so on.

This motion begins to “hone in” on a single limiting

location.) We see, for example, that the series

converges. Unfortunately,

the alternating series test gives us no indication as to

what the value of the sum could be. Generally, finding

the limit value is a considerable amount of work, if at

all possible. The values of many “simple” alternating

series are not known today. (One can show, however,

that the above series above converges to π2/12. See

CONVERGENT SERIES

.)

See also

ZETA FUNCTION

.

altitude A line segment indicating the height of a

two- or three-dimensional geometric figure such as a

POLYGON

,

POLYHEDRON

,

CYLINDER

, or

CONE

is called

an altitude of the figure. An altitude meets the base of

the figure at a

RIGHT ANGLE

.

Any

TRIANGLE

has three distinct altitudes. Each is a

LINE

segment emanating from a vertex of the triangle

meeting the opposite edge at a 90°angle. The

LAW OF

SINES

shows that the lengths ha, hb, and hcof the three

altitudes of a triangle ABC satisfy:

ha= csin β= bsin γ

hb= asin γ= csin α

hc= bsin α= asin β

where a, b, and care the side-lengths of the triangle,

and α, β, and γare the angles at vertices A, B, and C,

respectively. Here hais the altitude meeting the side of

length aat 90°. Similarly, hband hcare the altitudes

meeting sides of length band c, respectively. It also fol-

lows from this law that the following relation holds:

where ris the radius of the circle that contains the

points A, B, and C.

The three altitudes of a triangle always meet at a

common point called the orthocenter of the triangle.

Surprisingly, this fundamental fact was not noticed by

the geometer E

UCLID

(ca. 300

B

.

C

.

E

.). The claim can be

proved as follows:

Given a triangle ABC, draw three lines, one

through each vertex and parallel to the side

opposite to that vertex. This creates a larger

triangle DEF.

By the

PARALLEL POSTULATE

, alternate

angles across parallel lines are equal. This

allows us to establish that all the angles in the

diagram have the values as shown. Conse-

quently, triangle DAB is similar to triangle

ABC and, in fact, is congruent to it, since it

shares the common side AB. We have that DA

is the same length as BC. In a similar way we

can show that AE also has the same length as

BC, and so Ais the midpoint of side DE of the

large triangle. Similarly, Bis the midpoint of

side DF, and Cthe midpoint of side EF. The

study of

EQUIDISTANT

points establishes that