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I read this in a maths magazine once, but never quite got the answer.

You are given a right-angled isosceles triangle. You need to draw in straight lines to form acute triangles, until you are left with only acute angled triangles. (By triangles I am not counting triangles that can be formed by two or more adjacent small triangles together.)

I read that the answer is that you'll be left with 7 acute-angled triangles.

Can anyone prove this to me?


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up vote 2 down vote accepted

You can do it with $7$ triangles by putting a point $P$ inside the triangle, near the right angle, and then drawing $5$ line segments from it, one to the right angle, one to each leg, and two to the hypotenuse, and then drawing a line segment connecting each leg point to its corresponding hypotenuse point. How to prove it can't be done with fewer than $7$ triangles, I don't see. If you have a way to search the Mathematical Games columns of Martin Gardner, I have a hunch you'll find it there.

[I don't think the triangle has to be isosceles, so long as it is right-angled]

EDIT: Ha! See page 58 of Gardner's book, My Best Mathematical and Logic Puzzles. Google may find it for you if you search for "an obtuse triangle cannot be dissected".

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Thanks Gerry Myerson! Good answer – mathsnoob Feb 24 '13 at 6:26

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