Analytic Foundations of Geometry

Robert S. Wilson

3. Equations of Circles

Theorems 3.1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Theorem 3.1: The equation of the circle whose center is   (x0, y0)   and whose radius is   r   is

(x - x0)2 + (y - y0)2 = r 2

Theorem 3.2: The circle whose equation is

(x - x0)2 + (y - y0)2 = r 2

and the vertical line   x = a   meet at the points where

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Theorem 3.3: The circle whose equation is

(x - x0)2 + (y - y0)2 = r 2

and the line

y = mx + b

meet at the points where

and

or

and

 

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Theorem 3.4: (Pythagoras)1 Let   f,   d,   and   r   be the three sides of a right triangle,   r   the hypoteneuse.

Then

a2 + f2 = r2

 

Theorem 3.5: (The Triangle Inequality) Let   A,   B,   and   C   be three points in the plane. Then the distance from   A   to   B   is less than or equal to the sum of the distances from   A   to   C   and   B   to   C   with equality if and only if   C   is on the line segment between   A   and   B.

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Theorem 3.6: The shortest distance from a point to a line is the perpendicular distance.

Theorem 3.7: A tangent line to a circle is perpendicular to the radius to the point of tangency.

Theorem 3.8: If a line is tangent to a circle, then all of the points which are either on the circle or inside the circle except for the point of tangency are all on the same side of the line.

Theorem 3.9: Let   A   and   B   be two points on a circle. The foot of the center of the circle in the line determined by   A   and   B   is the midpoint of the line segment between   A   and   B.

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Theorem 3.10: The points which lie on the intersection of two circles which have different centers whose equations are

(x - x1)2 + (y - y1)2 = r12

and

(x - x2)2 + (y - y2)2 = r22

lie on the line whose equation is

if the line joining the centers is horizontal and

otherwise

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Theorem 3.11: The line joining the two points where two circles intersect is perpendicular to the line joining their centers.

Theorem 3.12: The point where the line joining the centers of two circles meets the line containing the points of intersection of the two circles is given by

and

where   (x1, y1)   and   (x2, y2)   are the centers of the circles,   r1   and   r2   are the respective radii, and   d   is the distance between the centers.2

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Theorem 3.13: Given two circles whose equations are

(x - x1)2 + (y - y1)2 = r12

and

(x - x2)2 + (y - y2)2 = r22

their points of intersection are given by3

and

where

Theorem 3.14: If two circles have the same radius, then the points of intersection between the two circles lie on the perpendicular bisector of the line segment joining the two centers.4

Theorem 3.15: Given a circle with center   (x0, y0)   and radius   r   and a point   (x1, y1),

Theorem 3.16: Given a point   A   not on a line, any point whose distance from   A   is less than the distance from   A   to its foot in the line is on the same side of the line as   A.

 

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4. Translations and Reflections