$$\begin{align*} \cos\angle C & =\frac{e}{a}\\ e & =a\cos\angle C\\ f & =b-e\\ & =b-a\cos\angle C\\ \sin\angle C & =\frac{d}{a}\\ d & =a\sin\angle C\\ c^{2} & =f^{2}+d^{2}\\ & =(b-a\cos\angle C)^{2}+(a\sin\angle C)^{2}\\ & =b^{2}-2ab\cos\angle C+a^{2}\cos^{2}\angle C+a\sin^{2}\angle C\\ & =a^{2}(\cos^{2}\angle C+\sin^{2}\angle C)+b^{2}-2ab\cos\angle C\\ & =a^{2}+b^{2}-2ab\cos\angle C \end{align*}$$