# 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 θ, θ, …, θ. When , the distances are θ, θ, θ, θ, …, θ.

θ<π/4

0

tan

1

tan

n-1

tan

θ>π/4

1

tan

0

tan

1

cot

2

cot

n-2

cot