Recursive Similar Triangles

Start with a right triangle with right angle at the origin, horizontal base 1 and angle at the bottom-right corner, with . Label the vertices on the hypotenuse and ; when , the point is on the axis and is on the axis, and when , they are reversed.
This Demonstration iteratively draws similar triangles within the original triangle by adding vertices , , …, alternately on the and axes; all points with even indices are on one axis and all points with odd indices are on the other.
When , the distances of the vertices from the origin are , , …, . When , the distances are , , , , …, .

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