This shows the method of solving a quadratic using a ruler-and-compass construction. Drag the point "A" around on the screen to modify the quadratic and thus its roots.
The construction works by virtue of two cyclic quadrilaterals related to the quadratic function under consideration. One is an isosceles trapezium built from the roots and the y-intercept. The other is built from four triangles which are pairwise similar. These quadrilaterals share three vertices and so their circumcircles coincide. By considering the trapezium, the roots are seen as the intersection of this circle and the x-axis. By considering the other quadrilateral, it is clear how to construct this circle using a ruler and compass construction.