# Recursive Similar Triangles

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.

θ

0<θ<π/2

P

-1

P

0

θ<π/4

P

-1

x

P

0

y

θ>π/4

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.

n

P

1

P

2

P

n

x

y

When , the distances of the vertices from the origin are , , …, . When , the distances are , , , , …, .

θ<π/4

tanθ

0

tanθ

1

tanθ

n-1

θ>π/4

tanθ

1

tanθ

0

cotθ

1

cotθ

2

cotθ

n-2