# What is the probability that a inscribed triangle in a circle lies in a semi-circle?

9 years ago
Method 1: Fix one endpoint of the chord, say at a vertex of the equilateral triangle. Randomly
choose a point on the circle for the other endpoint. The triangle divides the circle into three equal
parts; only points on the far side will yield chords longer than an edge of the triangle, so the
probability is 1/3.
Method 2: Consider a radius of the circle, say one that is perpendicular to one of the triangle’s
edges. Randomly choose a point on this radius, and construct the chord through that point that is
perpendicular to the radius. The triangle’s edge bisects the radius, so the probability that the chord
is longer than an edge of the triangle is 1/2.
Method 3: Randomly choose a point on the interior of the circle, and construct the chord that has
this point as its midpoint. The chord will be longer than an edge of the triangle if the point lies
inside a circle of half the radius of the larger circle, which occurs with probability 1/4

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