Ellipse Properties: Eccentricity, Tangents and Focal Radius

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2018-10-14 19:25

This GeoGebra demonstration illustrates the fundamental geometric properties of ellipses. It covers: (1) The definition of eccentricity e=c/a and its geometric meaning; (2) The focal radius property showing the sum of distances from any point on the ellipse to the two foci is constant; (3) The tangent line properties and the optical reflection characteristic. Construction uses commands: Eccentricity for computing eccentricity, Tangent for drawing tangents, OrthogonalLine for normals, Intersect for finding intersections, Angle for angle measurement, and Segment for drawing focal radius segments. Suitable for high school coordinate geometry courses and competition training.

GeoGebra Commands & Parameters

Eccentricity(c) → e_1
Point(c) → E
Tangent(E, c) → n,
OrthogonalLine(C, n) → q
Intersect(q, n) → F
Angle(E, F, C) → α
Segment(C, F) → s
Point(c) → A
Tangent(A, c) → h,
Intersect(p, h) → J, K
OrthogonalLine(K, h) → i
Intersect(c, i) → L, N
Intersect(h, i) → M
Angle(A, K, L) → γ
Intersect(g, l) → I
Intersect(f, l) → P
Point(l) → Q

AI Drawing Prompt Reference

Draw an ellipse, label its foci and major axis, and calculate the eccentricity. From any point on the ellipse, draw tangent lines and show the perpendicular relationship between tangents and focal radii. Also verify that the sum of distances from any point on the ellipse to the two foci is constant and demonstrate the optical reflection property of tangent lines.

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