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# Triangles inside circles

Best Answer bonanova, 25 March 2014 - 08:28 AM

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### #1 bonanova

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Posted 28 February 2014 - 12:18 AM

A few puzzles posted in this forum have related to random triangles inside a circle.

By evaluating nasty integrals, or by my preferred method, simulation, it can be shown, perhaps surprisingly, that triangles constructed from sets of three uniformly chosen points within a circle cover only about 7.388% of the circle's area on average. After looking at Phil's recent problem on the subject, I simulated 1 million triangles to determine the median area. It turns out to be about 5.335% of the circle's area. Read: a random triangle has a 50% chance of being smaller.

If the distribution of random-triangle areas has a mean of about 7.4% and a median of about 5.3%, what value might you expect for the mode?

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### #2 bonanova

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Posted 07 March 2014 - 03:07 PM

Spoiler for Hint

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Posted 14 March 2014 - 05:50 PM

Spoiler for guess

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### #4 bonanova

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Posted 15 March 2014 - 02:05 PM

It's lower than that.

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### #5 bonanova

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Posted 17 March 2014 - 12:30 AM

Spoiler for Big Hint

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### #6 bonanova

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Posted 25 March 2014 - 08:28 AM   Best Answer

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