MathDB

The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that
[*]all colored quadrilaterals are trapezoids; [*]the areas of all polygons of the same color are equal; [*]the ratios of the bases of one-color trapezoids are equal; [*]if

S_0=1

is the area of the original triangle, and

S_i

is the area of the colored polygons at the

i^{\text{th}}

step, then

S_1=1

S_2=5

and for

n\geqslant 3

the equality

S_n=5S_{n-1}-S_{n-2}

is satisfied.
Proposed by F. Nilov [img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png

Problem Statement