Question

three points are chosen uniformly at random on a circle. what is the probability that no two of these points form an obtuse triangle with the circle's center?

Answer 1

The probability that no two of these points form an obtuse triangle with the circle's center is $\frac{3}{4}$.

What is thales theorem?

According to Thales's Theorem, the angle ABC is a right angle if A, B, and C are distinct points on a circle where the line AC is a diameter. The 31st assertion in the third book of Euclid's Elements, which is a special case of the inscribed angle theorem, mentions and proves Thales's theorem. Although Pythagoras has also been given credit for it, Thales of Miletus is usually thought to be the author.

Calculation:

Let's start with Thales Theorem, which states that straight angles are those that subtend a diameter. If AB is a diameter, we will always have a right angle C with points ABC on a circle.

Let's assume that we never acquire a diameter because the right angle is a specific case between acute and obtuse that has no bearing on probability.

Assume that AB is one side of our triangle and that it is a random chord on the circle rather than a diameter. Antipodes A' and B', opposite locations where AA' and BB' are diameters, exist between A and B.

There are two arcs, one small and one long, in the pattern AB. So that ASB and A'S'B are the short arcs, shorter than a semicircle, let's place point S on the short side and L on the long side.

There are two arcs, one small and one long, in the pattern AB. So that ASB and A'S'B are the short arcs, shorter than a semicircle, let's place point S on the short side and L on the long side.

The crucial concept is: If our third point C is on the arc A'S'B, we obtain an acute triangle; if it is on the arc A'L'B, we obtain an obtuse triangle.

It should be clear that arc A'S'B', congruent to arc ASB, has an equal chance of being between 0 and 90° and between 90° and 180°. Because of the symmetry, we can infer that the average arc A'S'B' is 90°.

C is three times more likely to land on the long arc A'L'B', measuring 270° than the short arc A'S'B', measuring 90°, for that average arc. Thus, an obtuse triangle is three times more likely to exist.

Hence, The probability that no two of these points form an obtuse triangle with the circle's center is 3/4.

Learn more about thales theorem, here:

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