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Archimedes' Quadrature of Parabola Proposition 3

Archimedes' Quadrature of Parabola Proposition 3
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2015-02-12 10:04
FunctionCoordinate GeometryGeometric Construction
This example illustrates Archimedes' third proposition on the quadrature of the parabola, involving the properties and area calculation of parabolas. Using GeoGebra commands such as Point, Line, Parabola, Intersect, etc., it constructs points, lines, intersections, and tangents on the parabola, demonstrating how to solve the area under the parabola using geometric construction methods. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Line(A, B) → a
  3. Parabola(C, a) → c
  4. Line(A, C) → b
  5. Point(c) → P
  6. Line(P, b) → d
  7. Intersect(c, b, 1) → E
  8. Tangent(P, c) → e,
  9. Point(b) → G
  10. Point(c) → Q'
  11. Line(Q', e) → i
  12. Intersect(d, i) → V'
  13. Segment(Q', V') → k
  14. Point(c) → Q
  15. Line(Q, e) → f
  16. Intersect(d, f) → V
  17. Segment(Q, V) → g
  18. Polygon(Q, V, 4) → poly1, h, j, l, m, D, F
  19. Polygon(Q', V', 4) → poly2, n, p, q, r, H, I
  20. PointIn(poly1) → Pointpoly1
  21. PointIn(poly2) → Pointpoly2
  22. Segment(P, V) → s
  23. Segment(P, V') → t

AI Drawing Prompt Reference

Draw a parabola and mark a point on it. From this point, draw a tangent to the parabola and a perpendicular line to the parabola. Mark a point on the perpendicular line and draw a line through this point and another point on the parabola. Find the intersection points of the parabola with these two lines and connect them to form two quadrilaterals. Then, mark two points on the parabola, draw lines through these points and the tangent, find their intersections, and connect them to form two more quadrilaterals.
The prompt above was compiled from the original drawing commands and is for reference only.