Somewhat simplified, this proposition asserts that if ABC is any kind of triangle, and MC, NC are parallelograms on AC, BC, the opposite sides being produced to meet at P; and if PC is produced making QR = PC; and if the parallelogram AT is constructed, then AT = MC + NC.
For MC = AP = AR, having equal bases and equal altitudes.
Similarly, NC = QT.
Adding, MC + NC = AT.
If, now, ABC is a right triangle, and if MC and NC are squares, it is easy to show that AT is a square, and the proposition reduces to the Pythagorean Theorem.
The Arab writer, Al-Nairīzī (died about 922 A.D.), attributes to Thābit ben Qurra (826-901 A.D.) a proof substantially as follows:
The four triangles T can be proved congruent. Then if we take from the whole figure T and T', we have left the squares on the two sides of the right angle. If we take away the other two triangles instead, we have left the square on the hypotenuse. Therefore the former is equivalent to the latter.
A proof attributed to the great artist, Leonardo da Vinci (1452-1519), is as follows:
