You have control over the location of point E along the x-axis from 1 TO 4. Point N rotates around point E on a circle of radius 1. The blue circle always contains the origin and is centered at point N. The blue dotted line is tangent to the green circle (that N rotates on). The red dotted line is the perpendicular from the origin to the blue dotted tangent line. The point of intersection of those two lines is traced.
Although more complicated than some of my other sketches, it still shows what can be created WITHOUT USING TRIGONOMETRY that, in our textbooks, is usually not even mentioned until well after trig is covered in detail.
For the full file, visit http://tube.geogebra.org/material/show/id/266203