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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.

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.

Circles Touch Externally
Circles Touch Internally

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





10th Grade Math

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