Projection Formula

Theorem 3.4 (Projection Formula). In $\triangle ABC$,

Proof. Drop the altitude from $A$ to line $BC$, meeting it at $D$, so $a=BC=BD+DC$. In right triangle $ABD$, $\dfrac{BD}{AB}=\cos B\Rightarrow BD=c\cos B$; in right triangle $ADC$, $\dfrac{DC}{AC}=\cos C\Rightarrow DC=b\cos C$. Hence

The other two projection formulas follow the same way, dropping the altitude from $B$ or from $C$ instead. $\blacksquare$

Interpretation. $b\cos C$ is exactly the length of the projection of side $b$ onto side $a$, and $c\cos B$ is the projection of side $c$ onto side $a$. Geometrically: a side of a triangle equals the sum of the projections of the other two sides on it. (If $B$ or $C$ happens to be obtuse, the corresponding cosine is negative and that projection is counted as a negative length — the algebra automatically takes care of the geometry.) …