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No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of ...
A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line. A chord is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line ...
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: = + ′ ().
The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x 1, y 1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x 1, y 1), so it has the form (x 1 − a)x + (y 1 – b)y = c.
The secants ′ ¯, ′ ¯ meet on the radical axis of the given two circles. Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles. The center of the tangent circle is the intercept of the lines ¯, ¯. The secants ′ ¯, ′ ¯ become tangents at the points ,.
The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: = | | | | = | | | | Next to the intersecting chords theorem and the tangent-secant theorem , the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle ...