Given: ​CD¯¯¯¯¯​ is an altitude of △ABC. Prove: a2=b2+c2−2bccosA
Figure shows triangle A B C. Segment A B is the base and contains point D. Segment C D is shown forming a right angle. Segment C D is labeled h. Segment A B is labeled c. Segment B C is labeled a. Segment A C is labeled b. Segment A D is labeled x. Segment D B is labeled c minus x.

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Statement Reason
​CD¯¯¯¯¯​ is an altitude of △ABC. Given
​△ACD​ and ​△BCD​ are right triangles. Definition of right triangle
a2=(c−x)2+h2
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a2=c2−2cx+x2+h2 Square the binomial.
b2=x2+h2
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cosA=xb
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bcosA=x Multiplication Property of Equality
a2=c2−2c(bcosA)+b2
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​a2=b2+c2−2bccosA​ Commutative Properties of Addition and Multiplication