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lies on a line segment 
between 
and 
and circles are drawn having 
and 
as diameters. A common tangent to both circles touches the circle with as diameter at 
and the circle with as diameter at 
.
and the common tangent to both circles at all meet at a single point which lies on the circumference of the circle with as diameter.
be a triangle with 
and 
.
inside triangle satisfies the following conditions:![\[
XA^2 + BC^2 \leq XC^2 + AB^2 \leq XB^2 + AC^2.
\]](http://latex.artofproblemsolving.com/0/3/1/031aa58d7d3dabe636dfb4245ef2456a2a3d5df6.png)
Find the maximum value of 
such that![\[
\frac{XA}{AC} \geq m.
\]](http://latex.artofproblemsolving.com/c/5/e/c5ed2892b224aff3b2f3e20b4c9424707ff88690.png)
