Since Heath noted that Hippocrates was initially “circumscribing a semicircle about an isosceles right-angled triangle”, it is necessary to begin by constructing an isosceles right-angled triangle as shown in figure 14.
Figure 14: Isosceles right triangle ABC circumscribed by semicircle ABC.
An isosceles-right triangle could also have been constructed using points A, C, and D (see Figure 15). Therefore, AB = BC = CD = DA.
Figure 15: Construction of isosceles right triangle ABC.
Next, we construct a second circle with center D and radius DA to get “a segment of a circle similar to those cut off by the sides” of the isosceles right triangle ABC (see Figure 16). This segment of a circle with base AEC is in a quadrant of a circle and is therefore similar to the two original segments with base AB and BC.
Figure 16: Construction of a circle with center D.