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Consider △ABC with an altitude drawn from point B to AC¯¯¯¯¯.

Prove a2=c2+b2−2bccosA .

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The altitude creates two right triangles. By the Pythagorean theorem, a2=h2+(b−x)2, which becomes [tex]a^2=h^2+b^2- ??+x^2[/tex] after expanding the binomial.

Rearrange the terms to get [tex]a^2=??+h^2+b^2-??[/tex].

Using the Pythagorean theorem again, x² + h² = c², so by substitution, the previous equation becomes [tex]a^2=??+b^2-??[/tex].

Using right triangle trigonometry, [tex]cosA=\frac{??}{??}[/tex], so [tex]x=??cos??[/tex].

Performing another substitution gives the equation a2=c2+b2−2bccosA.