The Triangle Proportionality Theorem and similarity in right triangles

The Triangle Proportionality Theorem says that a line parallel to one side of a triangle, crossing the other two sides, divides those sides proportionally; its converse is also true. If a line parallel to side BC cuts AB at D and AC at E, with AD = 4, DB = 6, and AE = 6, then AD/DB = AE/EC gives 4/6 = 6/EC, so EC = 9. In a right triangle, the altitude drawn from the right angle to the hypotenuse creates two smaller triangles that are similar to the original triangle and to each other. That similarity gives geometric mean relationships: the altitude is the geometric mean of the two hypotenuse segments, so segments of 4 and 9 give an altitude of the square root of 36, or 6, and each leg is the geometric mean of the whole hypotenuse and the segment next to it. Adding the two leg relationships, a^2 = cx and b^2 = cy, gives a^2 + b^2 = c(x + y) = c^2, one proof of the Pythagorean theorem.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report