# Two Circles Touch each Other

Here we will prove that if two circles touch each other, the point of contact lies on the straight line joining their centres.

Case 1: When the two circles touch each other externally.

Given: Two circles with centres O and P touch each other externally at T.

To prove: T lies on the line OP.

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

 Statement Reason 1. ∠OTX = 90° 1. Radius OT ⊥ tangent XY. 2. ∠PTX = 90° 2. Radius PT ⊥ tangent XY. 3. ∠OTX + ∠PTX = 180°⟹ ∠OTP = 180°⟹ OTP is a straight line ⟹ T lies on OP. (Proved) 3. Adding statement 1 and 2.

Case 2: When the two circles touch each other internally at T.

To prove: T lies on OP produced.

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

 Statement Reason 1. ∠OTX = 90° 1. Radius OT ⊥ tangent XY. 2. ∠PTX = 90° 2. Radius PT ⊥ tangent XY. 3. OT and PT are both ⊥ to XY at the same point T. 3. From statement 1 and 2. 4. OT and PT lies on the same straight line⟹ OTP is a straight line⟹ T lies on OP. (Proved) 4. Only one perpendicular can be drawn to a line through a point on it.

Note: Let two circles with centres O and P touch each other at T. Let OT = r1 and PT = r2 and r1 > r2.

Let the distance between their centres = OP = d.

It is clear from the figures that

• When the circles touch externally, d = r1 + r2.

• When the circles touch internally, d = r1 - r2.