Analytic Foundations of Geometry

Robert S. Wilson

2. Parametric Equations of Lines

Theorem 2.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Theorem 2.1: (The parametric representation of a line) Given two points   (x1, y1)   and   (x2, y2),   the point   (x, y)   is on the line determined by   (x1, y1)   and   (x2, y2)   if and only if there is a real number t such that

x = (1 - t)x1 + tx2,

and

y = (1 - t)y1 + ty2

Theorem 2.2: (The parametric form of the Ruler Axiom) Let t be a real number. Let A and B be two points. Let

A = (x1, y1)

and

B = (x2, y2)

Let   t   be a real number, and let

C = (1 - t)A + tB

= ( (1 - t)x1 + tx2, (1 - t)y1 + ty2 )

using vector addition and scalar multiplication of points. Then, the distance from   A   to   C,

|AC| = |t| |AB| ,

where   |AB|   is the distance from   A   to   B, and the distance from   C   to   B,

|CB| = |1 - t| |AB| .

Which is to say that, if   C   is a point on the line segment between   A   and   B,   that

|AB| = |AC| + |CB|

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Theorem 2.3: If   C   is on the line segment between   A   and   B   then

|AB| = |AC| + |CB|

If   C   is on the line determined by   A   and   B   but on the other side of   B   from   A   then

|AB| = |AC| - |CB|

If   C   is on the line determined by   A   and   B   but on the other side of   A   from   B,   then

|AB| = |BC| - |AC|

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Corollary: (The midpoint formula) Let   (x1, y1)   and   (x2, y2)   be two points. The midpoint between them has coordinates1

Theorem 2.4: If   C   is on the line segment between   A   and   B   then   A   and   B   are on opposite sides of   C.

Theorem 2.5: Let   A   be a point on the line determined by the equation   ax + by = c,   and let   B   be a point not on that line. Then the points on the line determined by   A   and   B   which are on the same side of   A   as   B   are on the same side of the line   ax + by = c   as   B,   and the points on the other side of   A from   B   on the line determined by   A   and   B   are on the other side of the line   ax + by = c.

Theorem 2.6: If two lines are parallel, then all of the points on one line lie on the same side of the other line.

Theorem 2.7: Given points   A   and   B   and a line whose equation is   ax + by = c,   where   A   is either on the line or on the same side of the line as   B,   every point on the line segment between   A   and   B   is on the same side of the line as  B.

Theorem 2.8: If a line segment contains points on both sides of another line, then the line must intersect the segment somewhere between its endpoints.

Theorem 2.9: Let   A,   B,   and   C   be three noncolinear points, let   D   be a point on the line segment strictly between   A   and   B,   and let   E   be a point on the line segment strictly between   A   and   C.   Then   DE   is parallel to   BC   if and only if there is a nonzero real number   t   such that

D = (1 - t)A + tB

and

E = (1 - t)A + tC

for the same value of   t.

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Theorem 2.10: Let   A,   B,   and   C   be three noncolinear points. If   D   is on the line through   A   which is parallel to   BC   then there is a real number   s   such that

D = A + s(C - B)

Theorem 2.11: (The parametric representation of a plane) Let   A,   B,   and   C be three noncolinear points. Let   D   be any point in the plane. Then there are real numbers   q,   r,   and   s   such that

q + r + s = 1

and

D = qA + rB + sC

Theorem 2.12: Let   A,   B,   and   C   be three noncolinear points, and let

D = qA + rB + sC

be a point in the plane, where

q + r + s = 1

Then   D   is on the same side of   BC   as   A   if and only if   q > 0.

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Theorem 2.13: (The First Pasch property) Let   A,   B,   and   C   be three noncolinear points. If a line going through   A   contains points in the angle between   AB   and   AC,   then that line intersects the line segment   BC.

Theorem 2.14: (The Second Pasch property) Let   A,   B,   and   C be three noncolinear points. If a line intersects the line segment   AB,   then the line will either intersect line segment   AC,   segment   BC,   or go through point   C.

 

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3. Equations of Circles