336. The Line of Centres · where touching circles meet

Try to break it

Drag P along the line of centres. The two circles expand and contract, and at every position the touch point P stays exactly on the line joining the two centres. If P slipped off this line, the circles would either overlap at two points or separate — touching demands collinearity of the centres and the touch point.

How you build it

Mark both centres, draw the touching circles, then draw the line through the centres — see it pass straight through the point of contact, so the two centres and the contact point are collinear.

• Mark the centre A.
• With centre A, draw the larger circle.
• Mark the centre B inside the first circle.
• With centre B, draw the smaller circle so it touches the first at the point of contact P.
• Draw the line through the two centres A and B — watch it pass straight through the point of contact P. So A, B and P are collinear.

The proof, step by step

Prove that the point of contact of two touching circles lies on the line joining their centres.

• Draw a common tangent PQ at the point of contact P.
• Join AP and BP. Both are radii to the tangent, so ∠APQ = 90° and ∠BPQ = 90°.
• Since both AP and BP are perpendicular to PQ at P, they must lie on the same straight line.
• Therefore, P lies on the line AB produced.

Worked example

Two circles with centres O and P touch each other internally at Q. If OQ = 9 cm and OP = 4 cm, find the radius of the second circle.

Since circles touch internally, OQ = OP + PQ. Radius of second circle PQ = OQ - OP = 9 - 4 = 5 cm.

• 3 cm
• 5 cm — correct
• 13 cm
• 4.5 cm