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