What is the maximum area of the triangle in which the points lie?
There are n points in a given plane. Any three of the points form an triangle of area
For a particular configuration of the points, suppose triangle is the triangle of least area in which all the points lie.
Over all configurations, find the maximum possible area of triangle .
The answer is 4.
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1 solution
Among all triples of points,we choose a triple X , Y , Z such that triangle X Y Z has the maximal area F .Obviously F ≤ 1 .Draw parallels to opposite sides through X , Y , Z .You get triangle X 1 Y 1 Z 1 with area 4 F ≤ 4 .We show that triangle X 1 Y 1 Z 1 contains all n points. Suppose there is a point P outside triangle X 1 Y 1 Z 1 .Then triangle X , Y , Z and p lie on different sides of at least one of the lines X 1 Y 1 , Y 1 Z 1 , Z 1 X 1 .From here derive a contradiction.
Hence the answer is 4 .