Question 1158588
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Let C be the center of the circle; let P be the point of intersection  of CG and the circle.<br>
The shortest distance from G to a point on the circle is the length of GP.<br>
Angle AGB is 60 degrees, so angle AGC is 30 degrees.<br>
Then triangle AGC is a 30-60-90 right triangle.<br>
AG=12 means AC=6 and CG=6*sqrt(3).<br>
CP is also 6; that makes GP 6*sqrt(3)-6.<br>
ANSWER: 6*sqrt(3)-6<br>