Unformatted text preview: show that the equation of the quadratrix in polar coordinates is πr sin θ = 2 aθ. (b) Show that if P is on the arc of the circle in the Frst quadrant, then the angle ∠ POA can be trisected as follows: let F be the intersection of OP with the quadratrix, and let FH be the vertical line segment to the xaxis. If F ′ is onethird the way from H to F along this segment, and F ′ L is a horizontal segment with L on the quadratrix, then show that ∠ LOA = 1 3 ∠ POA . (c) i. Show that if the quadratrix intersects OA at G , then OG = 2 a π . (You can use calculus here: in the proof by Dinostratus ( ca. 390320 BC ), it is done by contradiction, using only Euclidean geometry.) ii. Conclude from this that ⌢ CA : OA = OA : OG. iii. Show how, in light of this, we can construct a line segment equal in length to ⌢ CA . iv. Show that a rectangle with one side equal to twice this line segment and the other equal to a has the same area as the circle of radius a ....
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 Fall '08
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 Calculus, Conic Sections, Euclidean geometry, OA, Pappus, quadratrix

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