Recursive Similar Triangles

angle θ (radians)

0.6283

iterations

50

labels

Start with a right triangle with right angle at the origin, horizontal base 1 and angle

θ

at the bottom-right corner, with

0<θ<π/2

. Label the vertices on the hypotenuse

P

-1

and

P

0

; when

θ<π/4

, the point

P

-1

is on the

x

axis and

P

0

is on the

y

axis, and when

θ>π/4

, they are reversed.

This Demonstration iteratively draws

n

similar triangles within the original triangle by adding vertices

P

1

,

P

2

, …,

P

n

alternately on the

x

and

y

axes; all points with even indices are on one axis and all points with odd indices are on the other.

When

θ<π/4

, the distances of the vertices from the origin are

0

tan

θ

,

1

tan

θ

, …,

n-1

tan

θ

. When

θ>π/4

, the distances are

1

tan

θ

,

0

tan

θ

,

1

cot

θ

,

2

cot

θ

, …,

n-2

cot

θ

.