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:

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:

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.

10th Grade Math

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