When a small circle rolls without slipping inside a circle of twice its radius,
any point fixed to the circumference of the small circle moves in a straight line.
This construction is called a Tusa couple or Copernicus' theorem,
and is related to the Trammel of Archimedes.
In the interactive demonstration below:
O
is the center of the large circle.
P
is the center of the small circle.
A
is a point fixed to the circumference of the small circle, and moves in a straight line.
The motion can be described using the vectors:
OP
= (r cos(θ), r sin(θ))
PA
= (r cos(-θ), r sin(-θ))
where r is the radius of the small circle and the slider controls θ.
The spin triangle puzzle just uses this idea three times.
Points B and C are also fixed to the circumference of the small circle,
and therefore also move in straight lines.
To fully describe the motion,
let β be the angle ABC.
Then the angle AOC is also β (initially),
and the angle APC is 2β.
The vector PC is always the rotation of the vector PA through the angle 2β.
PC
= (r cos(-θ + 2β), r sin(-θ + 2β))
Similarly let γ be the angle ACB.
PB
= (r cos(-θ - 2γ), r sin(-θ - 2γ))
The fact that C moves in a straight line suggests a trigonometric identity.
The slope of the line OC is given both by the ratio of the coordinates of C
and by the tangent of β.
( r sin(θ) + r sin(-θ + 2β) ) / ( r cos(θ) + r cos(-θ + 2β) ) = tan(β)
