Parabola Geometric Property Demo

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2016-04-05 09:52

This example demonstrates the geometric property that a tangent line to a parabola forms equal angles with the parabola's axis and the segment from the vertex to the point of tangency. It involves core concepts like parabola, tangent, angles, and vectors. Constructed using GeoGebra commands such as Line, Parabola, Tangent, and Vector. Suitable for high school and above students to learn about parabolic geometry in analytical geometry.

GeoGebra Commands & Parameters

Line(A, B) → a
Parabola(C, a) → c
OrthogonalLine(C, a) → b
Point(c) → D
Tangent(D, c) → d,
Intersect(b, d) → E
Intersect(a, b) → F
Distance(E, F) → e
Circle(D, (0.8 * e)) → g
Segment(C, D) → h
Intersect(g, d) → I, J
Intersect(g, h, 1) → K
CircleSector(D, K, J) → k
Dilate(C, If[-0.5 < i ≤ 0.5, 0.5 - i, 0], D) → C'
Segment(C, C') → j
Dilate(D, If[1 < i ≤ 2, 2 - i, 0], E) → D'
Vector(D, D') → u
Translate(k, u) → k'
Vector(D, E) → v
Translate(K, v) → K'
Translate(I, v) → I'
Angle(I', E, K') → α
CircleArc(D, K, J) → f
Rotate(k', If[2 < i ≤ 3, ((-(i - 2)) * α), (-α)], E) → k''
Intersect(b, a) → G
Circle(G, 0.1) → p
Intersect(p, a, 2) → H
Intersect(p, b, 2) → L
Angle(H, G, L) → β

AI Drawing Prompt Reference

Draw a parabola with a tangent line to demonstrate that the tangent forms equal angles with the parabola's axis and the segment from the vertex to the point of tangency. Include the parabola, tangent, vertex, and point of tangency. Use circles and sectors to highlight angle relationships, then add two auxiliary line segments and rotate a sector to show the geometric property, labeling the key angles.

The prompt above was compiled from the original drawing commands and is for reference only.