# Extending compass and straightedge construction with pins and strings.

Construction, using only idealized compasses and a straightedges is called the classical construction.
There is a number of possible constructions, using these tools, for example dividing a line segment in $n$ equal parts, drawing a tangent of a circle, given a point on the circle etc. There is also a number of constructions, that are proven to be impossible to construct, using the means of classical construction, such as squaring a circle, trisecting an arbitrary angle etc.

Question: If we extend our set of tools to include pins and strings, like we for constructing an ellipse, how much more can we construct in a finite amount of steps?

Properties of pins and strings are defined such that:

• Your pencil can slide along the string or be bound to any point on the string.
• A string cannot be caught on another string.
• Strings cannot stretch.
• Strings and pins have zero width.
• Strings can be made to get caught on a pin or to move freely above the pin.

I think that although you can construct more shapes than you could before (ellipses etc.) the points at which these shapes intersect will be ones you could also construct with just a compass and straight edge. This is because all these shapes will have formulas in terms of the Euclidean distance $\sqrt{x^2+y^2}$ and so the coordinates of the points can be found by repeatedly solving quadratic equations, a process which can already be done with a straight edge and compass.