Question

If two circles touch each externally, prove that their point of contact and their centres are collinear.

Answer 1

Two circles with centres $A$ and $B$ touch each other at $P$ externally.
To prove: $A,B$ and $P$ are collinear.
Construction: Draw the common tangent at $P$. Join $AP$ and $BP$.
Proof: $\angle APQ={90}^o$ .....(i)
(Radius is perpendicular to the tangent)
$\angle BPQ={90}^o$ .....(ii)
(radius is perpendicular to the tangent)
Adding (i) and (ii), we get
$\angle APQ+\angle BPQ={90}^o+{90}^o$
$\Rightarrow \angle APB={180}^o$
$\Rightarrow$ $APB$ is a straight line
$\therefore$ $A,B$ and $P$ are collinear.