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The Carom Problem
Problem: Given points A and C not on a line, find the point B on the line so that the sum of the lengths  mBA and mCB is as small as possible.

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Solution: Reflect the point C across the line to give a point C'. The line segment AC' is the shortest distance between A and C'.
AC' will meet the line at a point, X.  The point X will solve the problem.

Justification. Consider B different from X.  The reflection of CB across the line gives the segment C'B.

mBA + m CB = mBA + m C'B > m C'A = mC'X + mXA = mCX  +  mXA.