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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.
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.
EOP.