Important Points About Tangents and Their Intersection with Circles

1. What is a Tangent?
   A tangent to a circle is a straight line that just touches the circle at one spot. This spot is called the point of tangency.

2. Right Angle Relationship:
   If you draw a line from the center of the circle to the point where the tangent touches, this line (called the radius) will always make a right angle with the tangent line.
   So, if ( O ) is the center of the circle and ( P ) is the point where the tangent touches, we can say:
   ( OP ) is perpendicular to the tangent line.

3. Types of Tangents for Two Circles:
   When we have two circles, there are two kinds of tangents:

   • External Tangents: These do not cross the line that connects the centers of the circles.
   • Internal Tangents: These do cross the line connecting the centers.

4. Length of Tangent Segments:
   If you have a point outside the circle, let's call it ( A ), and you draw two tangents to points ( B ) and ( C ) on the circle, the lengths of these tangents will be the same.
   So, ( AB ) is equal to ( AC ).

5. Angle Information:
   The angle formed between a tangent and a line (called a chord) drawn to the point of tangency is equal to the angle made by that chord in the other section of the circle.
   This is called the Tangent-Chord Theorem.

Knowing these points makes it easier to solve different geometry problems about circles and tangents.