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Infinite lines because a circle has infinite lines of symmetry.

Q: How many lines can be drawn through a given point on a circle that is tangent to the circle?

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The tangent line only touches the outside of a circle at one given point. So an outside line perpendicular to the circle's diameter at 90 degrees should do.

the circle is inscribed in the polygon :]

Not enough information has been given to find the tangent BC but it will be perpendicular or at right angles to the radius of the circle.

Draw a circle with centre O. draw a tangent PR touching circle at P. Draw QP perpendicular to RP at point P, Qp lies in the circle. Now, angle OPR = 90 degree (radius perpendicular to tangent) also angle QPR = 90 degree (given) Therefore angle OPR = angle QPR. This is possible only when O lies on QP. Hence, it is prooved that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Answer By- Rajendra Meena, Jaipur, India. email: rajendra.meena21@gmail.com

... touches each circle in exactly one point on each circle. given any two circles where none is entirely inside or inside and tangent to the other, there are at most four straight lines that are tangent to both circles.

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The tangent line only touches the outside of a circle at one given point. So an outside line perpendicular to the circle's diameter at 90 degrees should do.

A tangent is a straight line that touches the circumference of a circle at a given point

the circle is inscribed in the polygon :]

Not enough information has been given to find the tangent BC but it will be perpendicular or at right angles to the radius of the circle.

A circle's tangent is exactly the same as a triangle's tangent. If you look at a circle, you can make the radius the hypotenuse. Then make a vertical line from the point, and a horizontal line from the center. If you look, you have a triangle made inside the circle. This is why angles can be measured in radians, a unit that is derived from the circumference of a circle.-------------------------------------------------------------------------------------------By doing a little calculus, we find that the slope of the equation of a circle-the slope of the tangent line-is given by the tangent of an angle.AnswerEverything written above is correct, but doesn't have anything to do with tangents (in the circle sense of the word). Suppose you're given an angle theta. Draw a circle together with two radii, one horizontal and the other at an angle theta from the first one. (So far, this is the same as above.) Now draw the tangent to the circle at X, the point where the non-horizontal radius meets the circumference. Let Y be the point where this tangent meets the horizontal line through the centre. Then, assuming the radius is 1, tan(theta) is the distance XY, which is the length of part of the tangent.

Draw a circle with centre O. draw a tangent PR touching circle at P. Draw QP perpendicular to RP at point P, Qp lies in the circle. Now, angle OPR = 90 degree (radius perpendicular to tangent) also angle QPR = 90 degree (given) Therefore angle OPR = angle QPR. This is possible only when O lies on QP. Hence, it is prooved that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Answer By- Rajendra Meena, Jaipur, India. email: rajendra.meena21@gmail.com

... touches each circle in exactly one point on each circle. given any two circles where none is entirely inside or inside and tangent to the other, there are at most four straight lines that are tangent to both circles.

Join the centre of the circle O and the point A .Extend it to both sides to form a line.This is the required locus

Tangent:In geometry, the tangent line (or simply the tangent) is a curve at a given point and is the straight line that "just touches" the curve at that point. As it passes through the point where the tangent line and the curve meet the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.Chord:A chord of a curve is a geometric line segment whose endpoints both lie on the outside of the circle.

be equidistant from the center of the circle. APEX!

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The curved line that traces the boundary or perimeter of a circle is called the circumference. A straight line that touches the circle at just one point is called a tangent. See the related link, 'The Parts of the Circle', below this answer.