Birkhoff's Axioms for Geometry
- line - A set of points.
- distance - The distance between any two points A
and B is a non-negative real number
d(A,B) such that
- angle - An angle is formed by three ordered points
A, O, B (A≠O,
B≠O: ∠AOB such that
m∠AOB is a real number (mod 2π)
- If A, B and C are distinct
points, we say that B is between points
A and C (A*B*C) if and only if
- Line Segment
- The points A and C together with all
points B between A and C
form the line segment AC.
- Half-Line; Endpoint
- The half-line m' with endpoint
O is defined by two points O,
A in line m (A≠O)
as the set of all points A' of m
such that O is not between A and
- If two distinct lines have no points in common they are
parallel. A line is always regarded as parallel
- Straight Angle; Right Angle; Perpendicular
- Two half-lines m, n through O
are said to form a straight angle if
m∠mOn=π. Two half-lines
m, n through O are said to
form a right angle if m∠mOn
= ±π/2, in which case we say that m
is perpendicular to n.
- Triangle; Vertices; Degenerate Triangle
- If A, B, C are three distinct
points the three segments AB, BC,
CA are said to form a triangle with sides
AB, BC, CA and vertices
A, B, C. If A,
B and C are collinear then triangle
ABC is said to be degenerate.
- Similar; Congruent
- Any two geometric figures are similar if there exists
a one-to-one correspondence between the points of the two figures
such that all corresponding distances are in proportion and
corresponding angles have equal measures (except, perhaps, for
their sign). Any two geometric figures are congruent
if they are similar with a constant of proportionality,
- Postulate I.
- Postulate of Line Measure.
The points A, B, ..., of any line can be put
into 1:1 correspondence with the real numbers x so that
d(A,B) for all points
A and B.
- Postulate II.
- Point-line Postulate.
One and only one line, l, contains any two distinct
points P and Q.
- Postulate III.
- Postulate of Angle Measure.
The half-lines (or rays) l, m, n, ...,
through any point O can be put into 1:1 correpondence
with the real numbers a (mod 2π) so that if
A and B are points (other than O)
of l and m, respectively, the difference
am - al
(mod 2π) of the numbers associated with lines
l and m is m∠AOB.
- Postulate IV.
- Postulate of Similarity.
If in two triangles ABC and A'B'C' and
for some constant k>0,
= kd(A, B),
= kd(A, C),
and m∠B'A'C' = m±∠BAC,
then also d(B', C')
= kd(B, C),
m∠C'B'A' = ±m∠CBA,
and m∠A'C'B' = ±m∠ACB.
- A Set of Postulates for Plane Geometry (Based on Scale and
Protractors), G. D. Birkhoff, Annals of Mathematics, 33, 1932
- Roads to Geometry, E. Wallace & S. West, Prentice-Hall (Sect 2.5 & App. C)
R. I. Campbell, firstname.lastname@example.org
10 Feb, 2002