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There is a pyramid with equilateral base and it is placed on a circle of infinite radius. The centre of the triangle and the circle coincide. What is the probability(or what it tends to) that we select a point on the circle and be able to see two faces of the pyramid.

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  • $\begingroup$ I'm assuming ordinary Euclidean geometry here. Then isn't a circle of infinite radius just a line? And what would be its center? $\endgroup$
    – MPW
    Apr 25, 2017 at 12:39
  • $\begingroup$ @mpw you have a point, but the question was framed this way $\endgroup$ Apr 25, 2017 at 15:03

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The lines extending the sides of the triangle partition the plane into a bounded region (the triangle) and six unbounded regions: three adjacent to the sides, and three adjacent to the vertices. To see a pair of sides you need to be in a region adjacent to a vertex. Each of these is V-shaped with angle $\frac{\pi}{3}$. Therefore at infinity they make up approximately half the circle, hence the probability is one half.

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    $\begingroup$ Nice. Seems that this argument could be used for any triangular base, since you have (at infinity, so to speak) three lines going through a point with the six regions alternating "see two sides" and "see one side". By vertical angles, the opposite angle regions are identical, so half the circle sees two sides and half sees one side. $\endgroup$
    – Ned
    Apr 25, 2017 at 15:13

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