Does anyone know any synthetic geometry theorems (so, no algebra at all) or sources with synthetic geometry theorems, that relate lengths to areas? My only reference currently is Euclid's Elements, and I was hoping I could find other theorems. It doesn't matter if it's very old.
If this is of interest of anyone, here are the ones I found in Euclid's Elements:
B1 P42: Generate a parallelogram equal in area to a given triangle.
B1 P45: Find a parallelogram equal in area to a given polygon
B1 P47: the pythagorean theorem
B2 P14: Find a square equal to any given polygon
B6 P1: areas of triangles and parallelograms of the same height are proportional to their bases.
B6 P16: rectangles of equal area have proportional sides
B12 P2: areas of circles are proportional to the squares on their diameters
I hope this list may be of help.
Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.
Boyer, Carl B., History of analytic geometry, Mineola, NY: Dover Publications (ISBN 0-486-43832-5). x, 291 p. (2004). ZBL1099.51002.
Mlodinow, Leonard, Euclid’s window. The story of geometry from parallel lines to hyperspace, New York, NY: Free Press. xii, 306 p. (2001). ZBL0990.01001.\
Halsted, G. B., Synthetic projective geometry. ((2^{\text{nd}}) edition)., New York: Wiley. 37 S. (8^\circ). [Mathematical monographs edited by M. Merriman aiid R. S. Woodward. Nr. 2.] (1906). ZBL37.0563.03.