Right Triangle in a Semicircle and the Mean Proportional
e is a right angle; 6 is the mean of 4 and 9
Referring back to the previous page ('it is concluded by the preceding'), this sheet inscribes a triangle in a semicircle so that the angle at e is right. The hypotenuse is divided at the foot of the altitude into the segments a 9 b and b 4 d, with the altitude e equal to 6 and a further segment c of 4. The note '4 times 9 is 36, and 6 times 6 is 36' shows that 6 is the mean proportional between 4 and 9, the semicircle construction of the geometric mean.
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Right angle in a semicircle
A triangle is inscribed in a semicircle and marked e - 6 - a 9 b 4 d, with the statement 'e is right.' The angle standing on the diameter is a right angle, the theorem attributed to Thales.
6 as the mean proportional of 4 and 9
The segment note reads c - 4 with the calculation '4 times 9 is 36, and 6 times 6 is 36.' Because the altitude squared equals the product of the two parts of the hypotenuse, 6 is the mean proportional between 4 and 9.