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
Theorem 3.2: The circle whose equation is
and the vertical line x = a meet at the points where
Theorem 3.3: The circle whose equation is
and the line
meet at the points where
and
or
and
Theorem 3.4: (Pythagoras)1 Let f, d, and r be the three sides of a right triangle, r the hypoteneuse.
Then
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.
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.
Theorem 3.10: The points which lie on the intersection of two circles which have different centers whose equations are
and
lie on the line whose equation is
if the line joining the centers is horizontal and
otherwise
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
Theorem 3.13: Given two circles whose equations are
and
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.
4. Translations and Reflections