Understanding this proof for $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ from Gelfand

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We would like to have a small question concerning this proof for $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ from Gelfand's Trigonometry

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Why is it true that $\frac{q}{d}=\frac{a}{d}$. Is it necessary to use properties of similar triangles?

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Not at all. You simply use the fact that $q=a$.