Question

Prove that: If two circles touch each other, then the point of contact will lie on the line joining the two centres.

Answer 1

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)

$\begin{array}{ccc}Step& Statement& Reason\\ 1& \angle APQ={90}^{\circ }=\angle BPQ& RQistangenttothecirclesatP,APandBPareradii\\ 2& \angle APQ+\angle BPQ={180}^{\circ }& Fromstep1\\ 3& APBisastraightline& Angles\angle APQand\angle BPQisalinearpair\\ \therefore A,BandParecollinear& & \end{array}$
Proof:
(When circles touch internally)

$\begin{array}{ccc}Step& Statement& Reason\\ 1& APandBPareperpendiculartosamelineRQ& RQistangenttothecirclesatP,APandBPareradii\\ 2& BisapointonlineAP& \\ 3& APBisastraightline& Step2\\ \therefore A,BandParecollinear& & \end{array}$