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A line a is drawn from O, the center of the circle, through the radial point T; The line t is the perpendicular line to a. Construction of a tangent to a given circle (black) from a given exterior point (P). Thales' theorem may be used to construct the tangent lines to a point P external to the circle C:
Orthogonal circles. In geometry, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle ). A straight line through a circle's center is orthogonal to it, and if straight lines are also considered as a kind of generalized circles, for instance in inversive ...
For convenience the point should not lie on the perpendicular of line AB through the circle center. Construct a line (in orange), passing through C, that is parallel to the red line AB. If point C is on the perpendicular of AB through the circle center, the parallel would be a tangent to the circle. The construction is possible by other means ...
In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point . The set of all lines through a point is called a pencil, and their common intersection is called the vertex of the pencil. In any affine space (including a Euclidean space) the set of lines parallel to a given line (sharing the same ...
Ultraparallel theorem. Poincaré disc model: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line. In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel . The ultraparallel theorem states that every pair of (distinct ...
The line through the foci is called the major axis, and the line perpendicular to it through the center is the minor axis. The major axis intersects the ellipse at two vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance a {\displaystyle a} to the center.
The diagonals are perpendicular. The two line segments connecting opposite points of tangency have equal lengths. One pair of opposite tangent lengths have equal lengths. The bimedians have equal lengths. The products of opposite sides are equal. The center of the incircle lies on the diagonal that is the axis of symmetry.
Radical axis. In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail: For two circles c1, c2 with centers M1, M2 and radii r1, r2 the powers of a ...