If a circle and a line are drawn on a paper, three things can possible happen:

(i) The line does not touches or cuts the circle. As you see in the figure besides that does not touch or cut the circle.

(ii) The line cuts the circle in two parts. The straight line which cuts the circle in two points is called the secant of the circle. In this case the line cuts the circle at two points and . is a chord.

(iii) The line touches the circle at only one point. The line that touches the circle at only one point is called tangent of the circle. The point at which the line touches the circle is called the point of contact. In this case, you see that the line touches the circle at point (one point only).

Theorems related to tangents:

Theorem 18: The tangent at any point of a circle and the radius through this point are perpendicular to each other.

Given: is a tangent. Point of contact is .

To prove:

Proof: (since is outside the circle)

Similarly, we can show that out of all possible line segments that could be drawn from to the line would be the shortest. Hence .  (Reason: The shortest line segment, drawn from a given point to a given line is perpendicular to this line)

Hence Proved

Corollary: If two tangents are drawn to a circle from an external point, then

(i) the lengths of the tangents to the circle are equal

(ii) the tangents will subtend equal angles at the center of the circle

(iii) tangents are equally inclined to the line joining the point and the center of the circle.

Given: A circle with center . and are two tangents to this circle from external point .

To Prove:

(i)

(ii)

(iii)

Proof:

Consider and

(radius of the same circle)

(Theorem 18)

is common

(by R.H.S postulate)

Since corresponding parts of the congruent triangles are equal, we get

(i)

(ii)

(iii)

Hence Proved

Theorem 19: If two circles touch each other, the point of contact lines on the straight line through the centers.

There are two possible scenarios

Case 1: when the circles just touch each other externally

Given: Two circles with centers and touch each other externally at point as shown in the diagram.

To Prove: lies on the line i.e. and are collinear.

Proof:

(angle between the radius and the tangent)

(angle between the radius and the tangent)

which means that is a straight line.

Hence Proved

Case 2: when the two circles touch each other internally

Given: Two circles with centers and touch each other internally at point as shown in the diagram.

To Prove:  lies on the line i.e. and are collinear.

Proof:

(angle between the radius and the tangent)

(angle between the radius and the tangent)

Therefore both and are perpendicular to the tangent at the point .

Therefore and lie on the same line because only one perpendicular can be drawn through a line through a point on it.

Hence Proved

Theorems related to Chords:

Theorem 20: If two chords of a circle intersect internally or externally then the product of the lengths of their segments is equal.

There are two possible cases.

Case 1: When the chords intersect internally

Given: Chords and of a circle intersect each other at point inside the circle.

To Prove:

Proof: Consider and

(angles in the same segment)

(angles in the same segment)

Therefore  (AAA Postulate)

(corresponding sides of similar triangles are proportional)

Hence Proved

Case 2: When the chords intersects externally

Given: Chords and of a circle, when produced,  intersect each other at point outside the circle.

To Prove:

Proof: Consider and

(external angle of a cyclic quadrilateral is equal to internal opposite angle)

 (external angle of a cyclic quadrilateral is equal to internal opposite angle)

(By AAA postulate)

(corresponding sides of similar triangles are proportional)

Hence Proved

Theorem 21: The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

Given: A circle with center . Tangent touches the circle at point . is a chord drawn through point .

To Prove:  and

Proof: In

 (angle in a semi circle)

(angle between the radius and the tangent)

But (angles in the same segment)

Now ( is a straight line)

(Opposite angles of a cyclic quadrilateral)

as

Hence Proved

Theorem 22: If a chord and a tangent intersect externally, then the product of the length of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.

Given: Chord and tangent of a circle intersect each other at point P outside the circle.

To Prove:

Proof: Consider and

(Angles in alternate segment)

(common angle)

(AAA Postulate)

(corresponding sides of similar triangles are proportional)

Hence Proved