Question

Prove that "If two circles touch each other externally then their centres and the point of contact are collinear"

Answer 1

Data : $A$ and $B$ are the centres of touching circles. $P$ is point of contact.
To prove : $A,P$ and $B$ are collinear
Construction : Draw the common tangent $XPY$
Proof : $\angle APX={90}^{\circ }$ (i) $(AP\perp XY)$
$\angle BPX={90}^{\circ }$ (ii) $(BP\perp XY)$
Add (i) and (ii)
$\angle APX+\angle BPX={180}^{\circ }$
$\angle APB={180}^{\circ }$
$APB$ is a straight line
$\therefore A,P,B$ are collinear.