Given: Right △ABC as shown where CD is an altitude of the triangle. We prove that a^2+b^2=c^2
Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA.
Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA.
The proportions c\a and a\f are true because they are ratios of corresponding parts of similar triangles.
The two proportions can be rewritten as a^2=cf and b^2=ce
Adding b^2 to both sides of first equation, a^2=cf , results in the equation a^2 + b^2 = cf+ b^2 .
Because b^2 and ce are equal, ce can be substituted into the right side of the equation for b^2 , resulting in the equation a^2 + b^2 = cf + ce .
Applying the converse of the distributive property results in the equation .
a^2 + b^2 = c (f + e)
The last sentence of the proof is
Because f + e = c, a^2 + b^2 = c^2 .
