How to square plywood using the 3-4-5 method?
This is not far off from your original description, but consider that a set of yard sticks of reliable precision will do what you require. Drill a small hole in each one at the 1/2" point (or 1" if you like) and again at the appropriate multiple plus the 1/2" or 1" offset. Place a brad at your zero point, which could be a salvage board under the edge of the primary surface and place another brad at the appropriate distance. You now have two arc-devices which will provide you with the third point when the brad-holes align at the ends.

If precision is required, use your salvage board with the two brads and tap the edge of the brad into the work surface until half the diameter is "buried" to compensate for the thickness of the brad.

You can use any material, of course, and drill holes to your specifications in order to fit the working surface.