TITLE: Reflection on a circle
QUESTION [6 upvotes]: Illustration of the problem http://imageshack.us/a/img405/561/fjf5.png
Given two points "A" and "B" outside of a given circle of center "O". Where is the point X on the circle, such that AX + XB is the shortest possible?
For the problem "Given two points "A" and "B" on the same side of a given line. Where is the point X on the line, such that AX + XB is the shortest possible", there is that trick of reflecting one of the points about the given line, let's say the point A, and then the point X is the intersection of the line A'B with the given line. But I did not find a way of using that trick to this variation that changes the line for a circle...
Is there a way of solving the problem without long calculations? Something like the trick for the points and a straight line?

REPLY [3 votes]: Another way to see the bisection property of user8268's answer: The set of points with $AX+XB=k$ is an ellipse with foci $A,B$. Increase $k$ until the least value for which there is a point $X$ on the circle and also on the ellipse. There is only one such point, since if there were two, $P_1,P_2$, the $k$ value could be decreased somewhat (using some point between $P_1,P_2$ on the ellipse to give the length sum $k'<k$). An ellipse has the property that the two segments from the foci to a point $X$ on the ellipse make equal angles to the normal to the ellipse at $X$, so since the ellipse is tangent to the circle at $X$ we have the desired reflection property (equal angles of incidence and reflection).
Note: The above construction depends not only on $A,B$ being outside the given circle, but also on the whole segment $AB$ being outside it, so that a small enough ellipse doesn't meet the circle. 
It's another matter, given the points $A,B$ and the circle, to construct the point $X$. Restated: Given points $A,B$ outside a circle, construct the (smaller) ellipse with foci $A,B$ and tangent to the circle. I've tried this geometrically and analytically with no success, and don't know whether in general it is a "constructible" point (ruler/compass).