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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.

$Two Circles Touch each Other Externally$

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.

$Two Circles Touch each Other Internally$

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.