Geometric reconstruction · after Micmaths, 16:37–19:10

Pandrosion in the rectangle

A rectangle of width 2 and height 4. Two families of parallel lines construct the nested triangles; a third transfers the correction.

Pandrosion telescoping constructionRectangle OABC, moving point P on the right side, three horizontal lines and parallel diagonals, followed by a green correction to P prime.
HorizontalsTelescoping parallelsCorrection transfer

Move P along the right side, or use the slider.
Length L₂B₂—
Vertical gap to MN—
Why the triangles are similar

O, B₁, B₂, B₃ and B are collinear. The bases CB, L₁B₁, L₂B₂ and L₃B₃ are horizontal. The segments L₁B, L₂B₁ and L₃B₂ are parallel.

The lengths CB, L₁B₁, L₂B₂ and L₃B₃ therefore form a geometric progression. When L₃ reaches M, the last length is 1: the four lengths are then 2, ∛4, ∛2, 1.

What is actually constructed

Each interior point is computed as the intersection of two lines from its dependencies. No cube root or iteration formula is used to place the points.

The rectangle, midpoints and parallels are standard straightedge-and-compass constructions. Parallels are used here as composite tools; their auxiliary arcs are not shown.