Eratosthenes' Mean Finder for Duplicating the Cube

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2015-03-20 04:25

This example illustrates Eratosthenes' mechanical method to find two mean proportional to two given numbers, equivalent to duplicating the cube. Using commands like Polygon, Ray, Segment, Point, Line, Intersect, OrthogonalLine, Distance, it constructs the geometric figure. Suitable for high school geometry education.

GeoGebra Commands & Parameters

Polygon(A, E, 4) → poly1, a, b, c, d, C_1, D_1
Ray(A, D_1) → e
Ray(E, C_1) → f
Segment(A, C_1) → g
Point(e) → M'
Line(M', g) → h
Point(e) → N'
Line(N', g) → i
Intersect(h, f) → G
Intersect(i, f) → H
Segment(M', G) → j
Segment(N', H) → k
OrthogonalLine(M', e) → l
OrthogonalLine(N', e) → m
OrthogonalLine(G, f) → q
OrthogonalLine(H, f) → r
Intersect(q, e) → N
Intersect(r, e) → Q
Segment(G, N) → s
Segment(H, Q) → t
Point(t) → D
Segment(A, D) → n
Intersect(j, n) → B
Intersect(c, n) → M_1
Intersect(k, n) → C
Intersect(s, n) → O
Distance(B, M_1) → d_1
Distance(C, O) → d_2
Segment(D, H) → p
Segment(G, C) → a_1
Segment(C_1, B) → b_1
Intersect(c, f) → F
Intersect(d, e) → I
Intersect(c, d) → M

AI Drawing Prompt Reference

Draw a geometric figure with points A and E and line segment AC1. Construct rays AD1 and EC1, find their intersections with line segment AC1 at points M' and N'. Then, construct perpendicular lines through M' and N' to rays AD1 and EC1, find intersections G and H. Construct line segments MG and NH, then construct parallel lines to AC1 through G and H, find intersections B, M1, C, and O. Finally, construct line segments BD, GC, CB, and FH, and find intersections F, I, and M.

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