Geometry

In triangle OAP, angle OAP = 90 degrees. Angle APO = 50/2 = 25 degrees. In triangle OAP, angle AOP = 180 - 90 - 25 = 65 degrees. In triangle OAB, OA = OB (radii), so angle OAB = angle OBA. Angle AOB = 360 - 2 * angle AOP = 360 - 130 = 230? No, angle AOB = 180 - 50 = 130. Angle OAB = (180 - 130)/2 = 25 degrees.

The tangents at A and B form a quadrilateral with the center. The angles between the radius and tangent are 90 degrees. If the central angle is 60, the angle between tangents is 180 - 60 = 120 degrees.

From any point outside a circle, exactly two tangents can be drawn to the circle.

By geometric theorem, the tangent to a circle is always perpendicular to the radius at the point of contact.

By definition, a tangent to a circle touches it at exactly one point. At any specific point on the circumference, there is only one unique tangent line.

A tangent is a line that touches a circle at exactly one point. It is a limiting case of a secant where the two intersection points coincide.

From any point on the outer circle, you can draw two tangents to the inner circle, provided the point is outside the inner circle.

The sum of the radii of the two circles is 3 cm + 3 cm = 6 cm. Since the distance between their centers is 8 cm, which is greater than the sum of their radii, the two circles lie completely outside each other without intersecting. For two such disjoint circles, there are exactly 4 common tangents: 2 direct and 2 transverse.

A secant is a line that intersects a circle at two points. From any point inside a circle, one can draw an infinite number of lines passing through that point, and each of those lines will intersect the circle at two distinct points.

In a quadrilateral formed by the center O and the tangents, the angles at the points of contact are 90 degrees. Thus, angle AOB + angle APB = 180 degrees. 180 - 80 = 100 degrees.

The radius, tangent, and line from center to intersection form a right triangle. Radius = 5, hypotenuse = 13. Tangent length = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12.

A line can be tangent to infinitely many circles (e.g., circles of varying radii with centers at varying distances from the line).

By the tangent-secant theorem, PC^2 = PA * PB. PA = AB + BP = 10 + 8 = 18. PB = 8. PC^2 = 18 * 8 = 144. PC = 12.

For two circles to have two common tangents, they must intersect at two distinct points. This occurs when the distance between their centers is less than the sum of their radii and greater than the absolute difference of their radii.