# Concave Random Quadrilaterals from Four Points in a Disk

Concave Random Quadrilaterals from Four Points in a Disk

Four random points are chosen inside a circle. A quadrilateral is formed using the points as its vertices. Is concave or convex?

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The probability that is concave is known to be . The plot shows the four points and the convex hull of the points; the convex hull is the smallest convex polygon that encloses all of the points. If has four vertices, is convex, but if has only three vertices, is concave and one of the fours points is inside .

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35/(12)=0.29552

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π

C

C

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C

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C

If is concave, it is not unique: the quadrilateral can be formed in three ways; in the concave case, we show and not .

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C

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