*luaghs for help*. ... help me lol

we have given that δ abc is similar to δ cbd and δ abc is similar to δ acd according to the attached picture.

because of the similarity the corresponding sides are proportional. then a/c = f/a

and b/e = c/b. if we cross multiply, we'll get

a² = cf and b² = ce

if we add these equalities together, we'll get

a² + b² = c (f + e)

from the beginning, we know that c = e+f

then, the final result is:

a² + b² = c²