Prove that: If two circles touch each other, then the point of contact will lie on the line joining the two centres.
Open in App
Solution
Theorem - If two circles touch each other internally or externally, the point of contact and the centres of the circles are collinear. Data: Two circles with centres A and B each other externally at point P (Fig 1) or internally (Fig 2). To prove: A, B and P are collinear Construction: Draw the common tangent RPQ at P. Join AP and BP Proof: (When circle touch externally) StepStatementReason1∠APQ=90∘=∠BPQRQistangenttothecirclesatP,APandBPareradii2∠APQ+∠BPQ=180∘Fromstep13APBisastraightlineAngles∠APQand∠BPQisalinearpair∴A,BandParecollinear Proof: (When circles touch internally) StepStatementReason1APandBPareperpendiculartosamelineRQRQistangenttothecirclesatP,APandBPareradii2BisapointonlineAP3APBisastraightlineStep2∴A,BandParecollinear