write the three cases why circle have infinite number tangents
Answers
Answer:
Because tangent can be drawn to each and every point to the circle
Step-by-step explanation:
In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.
Tangent lines to one circleEdit
A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed.
The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of circle and tangent line has a reflection symmetry about the axis of the radius.

By the power-of-a-point theorem, the product of lengths PM·PN for any ray PMN equals to the square of PT, the length of the tangent line segment (red).
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 the circle. Thus the lengths of the segments from P to the two tangent points are equal. By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P.

The angle θ between a chord and a tangent is half the arc belonging to the chord.
The tangent line t and the tangent point T have a conjugate relationship to one another, which has been generalized into the idea of pole points and polar lines. The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency.
If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and S, then ∠TPS and ∠TOS are supplementary (sum to 180°).
If a chord TM is drawn from the tangency point T of exterior point P and ∠PTM ≤ 90° then ∠PTM = (1/2)∠TOM.
Compass and straightedge constructionsEdit
It is relatively straightforward to construct a line t tangent to a circle at a point T on the circumference of the circle:
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:
A circle is drawn centered on the midpoint of the line segment OP, having diameter OP, where O is again the center of the circle C.
The intersection points T1 and T2 of the circle C and the new circle are the tangent points for lines passing through P, by the following argument.
The line segments OT1 and OT2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT1 and PT2, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T1 and T2 are tangent to the circle C.
Another method to construct the tangent lines to a point P external to the circle using only