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Parabola as an Envelope of Straight Lines

Parabola as an Envelope of Straight Lines
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2016-10-07 13:22
FunctionCoordinate GeometryGeometric Construction
This example demonstrates how to construct a parabola as an envelope of a family of straight lines. Key mathematical concepts include coordinate geometry of lines and parabolas. Using GeoGebra commands such as Line, Point, Slope, OrthogonalLine, Intersect, and Locus, one can dynamically explore the properties of the parabola. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Line(C, D) → m
  2. Point(m) → B
  3. Slope(m) → a
  4. OrthogonalLine(B, m) → h
  5. Intersect(p, h) → O
  6. Locus(O, B) → loc1
  7. Parabola(A, m) → f
  8. CurveCartesian(((t^(2) * a^(3)) + (t^(2) * a) + (((2 * t) * b) * a^(2)) + ((2 * t) * b) - (((2 * t) * a^(2)) * y_1) - ((2 * t) * y_1) + (b^(2) * a) - (((2 * b) * a) * y_1) + (a * x_1^(2)) + (a * y_1^(2))) / ((2 * b) + ((2 * a) * x_1) - (2 * y_1)), (((-t^(2)) * a^(2)) - t^(2) + (((2 * t) * a^(2)) * x_1) + ((2 * t) * x_1) + b^(2) + (((2 * b) * a) * x_1) - x_1^(2) - y_1^(2)) / ((2 * b) + ((2 * a) * x_1) - (2 * y_1)), t, -100, 100) → d
  9. OrthogonalLine(B, m) → i
  10. LineBisector(A, B) → j
  11. Intersect(j, i) → E