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# triangle in a circle probability

## Question

martin has a game in mind for gambling to confuse and confound. you start off with a circle, and you select 3 points on the circle such that the area of the triangle formed is at least half the circle.

simple enough. now he poses the question.

if the three points are selected randomly, what's the likelyhood of that being true?

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Very small (impossible.)

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What does mean "randomly"?

If I remember well, there is a controversy how to choose randomly points on a circle.

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What does mean "randomly"?

If I remember well, there is a controversy how to choose randomly points on a circle.

Uniform density of points on the circle.

Envision choosing x and y randomly along the sides of the enclosing square,

and discarding the point if it lies outside the circle.

Note that choosing a random point on a random radius favors picking points near the center.

There is controversy (generally resolved now) about what constitutes a random chord (Bertrand paradox.)

The usual resolution is to pick a random point on a random radius and draw the perpendicular chord.

Interestingly, what fails for getting a random point succeeds for getting a random chord.

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"you start off with a circle, and you select 3 points on the circle..."

This means you are not choosing any points of the *interior* of the circle. The likelihood has a probability of zero.

The largest triangle area-wise that can be formed with three points chosen *on* the circle is an equilateral one.

And it's area is less than half of the area of the circle.

If you want to use the word "on" an object for meaning picking random interior points, you better change the

object to a *disc* instead. A circle doesn't have any interior points to choose from, but a disc does.

That is, if you are told you are selecting points on a circle, then you are not selecting points in the interior

of the circle.

Edited by Perhaps check it again

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Excellent point.

"Within a circle" or "On a disk" are appropriate references to the location of geometrically interior points.

Most likely what Phil meant.

We would also use the complementary term "Ball" (solid) with "Sphere" (hollow.)

In a single dimension, "Line segment" (having an interior) and "Point pairs" (hollow) would be used.

Only in zero dimensions does a single term "Point" suffice.

What leads us astray at times is the fact that colloquial usage is not so strict.

A manhole cover can be called circular, for example, rather "disk-like," and everyone understands.

Hmmm...

I wonder what interior-inclusive, "disk-like" term we should use with "Square" and "Rectangle"?

Not to mention Pentagon, ...

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