12. 2. Proposition 12 of Book 2 of Euclid’s "Elements."

Obtuse triangle

If the perpendicular ad falls upon the extended line bc (according to Proposition 12 of Book 2), then I say that the square of the side bc In this diagram, bc is the side opposite the obtuse angle. which subtends the obtuse angle is greater than the two squares of the lines ab and ac that enclose that obtuse angle, by the amount of the rectangle formed by ad and bc In modern notation: a² = b² + c² + 2(c × d). This explains how to calculate the "excess" area when an angle is wider than a right angle. / ...the power of the lines in respect to these squares... The Latin "potens" refers to the geometric "power" or area-generating capacity of a line.

13. 2. Proposition 13 of Book 2 of Euclid’s "Elements."

In every triangle, the side which faces an acute angle is less than the two sides containing that acute angle, by twice the rectangle contained by one of those sides and the segment between the perpendicular and the acute angle, when the perpendicular falls within the figure.

Since in the right-angled triangle abc (figure a), the perpendicular ad descends to the base bc, therefore ad is the mean proportionalA value that relates to two others in a specific ratio (x:z as z:y). In geometry, the height of a right triangle is the mean proportional between the two segments of the base it divides. between bd and dc. Alternatively, ac is the mean proportional between bc and dc. Wherefore it is found that the rectangle of cb times db is equal to the square of ab, just as the rectangle of cb times dc is equal to the square of ac.

half of the excess of ad over ab

perpendicular, square, obtuse angle, acute angle, triangle, proportional, known chord, acute-angled original: oxigonium, right-angled original: orthogonium, obtuse-angled original: ambligonium