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how do you difine at random ? do you break the first one uniformly? and choose one of the sticks and break itat some uniformely random point? or the both points are coosen uniformely and indepedantly in the stick ?
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Alternative proof using the area A of the triangle with the three different orange heights h1, h2, h3. A=blue+pink+yellow A=1/2.side(h1+h2+h3) So: (h1+h2+h3)=2A This generalizes to any convex polygon with equal side lengths or in 3D to any polyhedron with faces of equal area.pic.twitter.com/vyNPGo6W3x
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I think you forget to divide by 'side', so that, h1+h2+h3 = (2A)/side = h where h denotes the hight of the equilateral triangle. It's a nice proof
, but is more analytical than geometrical
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that’s crazy
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which makes for a nice way of visualizing 3 quantities whose sum is fixed
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Now this strikes me as a profound insight. Gotta do some reading!
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There is an other way to prove that: the area of the equilateral sums up to the area o the 3 triangles that decomposse the equilateral: each one is defined with a side of the equilateral and the inside point: sum the 3 areas factorize by the equilateral side , and you get it
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Only if perpendicular to the sides
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That's what means distance.
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