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 $|BC|$. enter image description here

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.

  • $\begingroup$ Hint. Think about the angle the radii make with the common tangent at $O$. $\endgroup$ – 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 $AM$.

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  • $\begingroup$ How do you establish that the two circles have a common tangent line at $O$? $\endgroup$ – Dillon M Apr 11 '19 at 22:37

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