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<span>Given: Right △ABC as shown where CD is an altitude of the triangle. We prove that 
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
and 
are
true because they are ratios of corresponding parts of similar
triangles.
The two proportions can be rewritten as
and 
.
Adding
to both sides of first equation, 
, results in the
equation 
.
Because and ce are equal, ce can be
substituted into the right side of the equation for , resulting in the
equation 
.
Applying the converse of the distributive property results in the equation
.
The last sentence of the proof is
Because f + e = c,.</span>
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
The two proportions can be rewritten as
Adding
Because
Applying the converse of the distributive property results in the equation
The last sentence of the proof is
Because f + e = c,