Archimedes Squaring Of A Parabola

Archimedes, parabola, triangle, Archimedes triangle, theorem of Archimedes, Referat, Hausaufgabe, Archimedes Squaring Of A Parabola
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Speech Martin Wiesauer, Bernhard Engl Archimedes Squaring Of A Parabola To determine the area enclosed in a parabola section The squaring of a parabola is one of Archimedes most remarkable achievements. It was accomplished about 240 b.C. and is based upon the properties of Archimedes triangle. An Archimedes triangle is a triangle whose sides consist of two tangents to a parabola and the chord connecting the points of tangency. The last mentioned side is taken as the base line or the bas of the triangle. In order to construct such a triangle we draw the parallels to the parabola axis through the two points H and K of the directrix and erect the perpendicular bisectors upon the lines connecting H and K with the focus F. If we designate the point of intersection of the two perpendicular bisectors as S, the point of intersection of the first perpendicular bisector with the second parallel to the axis as B, then A and B are points of the parabola (classical construction of the parabola), and ASB is an Archimedes triangle. Since SA and SB are two perpendicular bisectors of the triangle FHK, the parallel to the axis through S is the third perpendicular bisector; it consequently passes through the center of HK, and, as the midline of the trapezoid AHKB, it also passes through the center M of AB. This gives us the theorem: The median to the base of an Archimedes triangle is a parallel to the axis. Let the parabola tangents through the point of intersection O of the median SM to the ...

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