5. The base of an acute triangle is of lesse power than the shankes are, by a double oblong made of one of the shankes, and the one segment of the same, from the said angle, unto the perpendicular of the toppe. 13 p. ij.
As in the triangle aei, let the angle at i, be taken for an acute angle. Here by the [4. e], two obongs of ei, and oi, with the quadrate of eo, are equall to the quadrates of ei, and oi. Let the quadrate of ao, be added to both in common. Here the quadrate of ei, with the quadrates of io, and oa, that is the [9 e xij], with the quadrate of ia, is equall to two oblongs of ei, and oi, with two quadrates of eo & oa, that is by the [9 e xij], with the quadrate of ea. Therefore two oblongs with the quadrate of the base, are equall to the quadrates of the shankes: And the base is exceeded of the shankes by two oblongs.
And from hence is had the segment of the shanke toward the angle, and by that the perpendicular in a triangle.
Therefore
6. If the square of the base of an acute angle be taken out of the squares of the shankes, the quotient of the halfe of the remaine, divided by the shanke, shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe.
As in the acute angled triangle aei, let the sides be 13, 20, 21. And let ae be the base of the acute angle. Now the quadrate or square of 13 the said base is 169: And the quadrate of 20, or ai, is 400: And of 21, or ei, is 441. The summe of which is 841. And 841, 169, are 672:
Whose halfe is 336. And the quotient of 336, divided by 21, is 16, the segment of the dividing shanke ei, from the angle aei, unto ao, the perpendicular of the toppe. Now 21, 16, are 5. Therefore the other segment or portion of the said ei, is 5.
Now againe from 169, the quadrate of the base 13, take 25, the quadrate of 5, the said segment: And the remaine shall be 144, for the quadrate of the perpendicular ao, by the [9 e xij].