Does the line connecting the centers of two externally tangent circles contain the point of tangency?

In the picture below, the two circles are externally tangent at point $O$; $BC$ is tangent to both circles; the radius of the large circle is $25$ and the radius of the small circle is $9$. Find $|BC|$. After some time trying to work out this problem myself, I've looked at the answer provided in the solution manual but still haven't been able to understand the following: the solution seems to take as a given that point $O$ lies on the line connecting the centers of both circles. Although it is intuitive to me, I wonder how to show that the point of tangency between the two circles sits on line $MA$? (This is the first time I've been exposed to the concept of externally tangent circles in the book, so I apologize if the answer to my question is obvious). Thanks.

• Hint. Think about the angle the radii make with the common tangent at $O$. – Ethan Bolker Feb 1 '18 at 19:26

Join $OM$ and $AO$ and construct tangent at $O$. Now let tangent at $O$ be called $OX$. So you have since radius is perpendicular to tangent, $OM\perp OX$ and $AO \perp OX$. Therefore $\angle MOX = \angle AOX = 90^\circ$. Since $\angle MOX + \angle AOX = 180^\circ$, $O$ lies in line $AM$.
• How do you establish that the two circles have a common tangent line at $O$? – Dillon M Apr 11 at 22:37