1. The problem of Apollonius of Perga. (Find a circler that touches the given three circles.) Calderon thought up the following approach. Think of each of the three circles as a cross-section of three cones by the same plane. The cones must have a 90 degree angle at the vertex. The three cones intersect
at a point. Construct a cone with the vertex at this point and the angle of 90 degrees tangent to each of the three previously constructed cones. Its intersection with the plane of the three circles solves Apollonius' problem.

Certainly it's impossile to use paper cones to completely follow the construction (what of the intersections of the cones.) But even with two cones it's very easy to visualize how the complete construction will work.

2. Archimedes who (according to legend) needed just a fulcrum to
move the Earth was a master of the (mechanical) Method wherewith
he replaced figures with their centers of gravity.

Prepare flat shapes to be hung by different points on a wall. Hang a shape and draw a vertical line through the point. Do this with another point. This will give a point of intersection of two lines. Now check that the line through any third point is concurrent with the first two at their point of intersection. I.e., all such lines pass through a single point.

Conclude that for triangles medians meet at a point.

Once the center of gravity has been found, the shapes can be balanced on the tip of a needle or a pointed stick.

Please let me know what you found.

All the best,
Alexander Bogomolny