Euclid lived around 300BC. Of the 13 Books of the Elements written, his was not the first, he wrote the 4th book. It was Hippocrates of Chios or Kos Island of Greece (born circa 470BC) who wrote the 1st Book. The 2nd and 3rd are missing.
We know from Euclid’s proof in Proposition V1.31 (Book 6, Proposition 31) that the sum of the square area of the hypotenuse is equal to the combined sum of the 2 smaller sides. He showed that even the sum of rectangles obeyed this law. This law is anchored in the familiar realization that the Pythagorean 3-4-5 Triangle proves that 3×3 + 4×4 = 5×5.
Begin with any semi-circle with diameter AB and point C on its circumference. Draw the triangle ABC. It was a well known geometrical fact that a triangle drawn in a semi-circle is always a right triangle, ie: it is 90 degrees. Draw 2 more semi-circles on the other 2 sides of the triangle.
The semi-circle on AB includes the shaded areas of parchment brown and red sections S1 and S2. Each of the other 2 smaller semi-circles include a region shaded both with red (S1 or S2) and grey (L1 or L2). Since ABC is a right triangle, the semi-circle on the hypotenuse is equal to the sum of the other 2 semi-circles.
As both the semi-circle on the hypotenuse and the sum of the other 2 semi-circles include the red regions, therefore, by subtracting the red regions, we can conclude logically that the 2 grey regions equal the area of the right-angled triangle shown in light parchment colour.
This is an astounding and ancient revelation, that the areas of straight-lined polygons equal the areas of curved moon-like geometries called lunes.
ie: The famous Lunes of Hippocrates is the first solved area between curved lines.