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Proof 9: Rhombus area

A rhombus actually consists of four right-angled triangles, because the diagonals $d_1$ and $d_2$ are perpendicular. Those triangles are pair-wise equal. The rhombus area is thus

$$\begin{align}
A_\text{rhombus}&= A_{\text{triangle }a}+ A_{\text{triangle }b} + A_{\text{triangle }c} + A_{\text{triangle }d}\\
~&= 2A_{\text{triangle }a} +2 A_{\text{triangle }b}\qquad\leftarrow \text{ Pairwise equal triangles}\\
~&=\cancel2\frac1{ \cancel 2}l_aw_a+ \cancel 2\frac1{ \cancel 2}l_bw_b \qquad\leftarrow \text{Triangle area, }A=\frac12lw\\
~&=l_aw_a+l_bw_b \\
~&=\underbrace{\frac12 d_1}_{l_a} \underbrace{ \frac12 d_2 }_{w_a} + \underbrace{ \frac12 d_1 }_{l_b} \underbrace{ \frac12 d_2 }_{w_b} \\
~&=\cancel 2\,\frac1{\cancel2} d_1\frac 12 d_2\\
~&=\frac12 d_1d_2\qquad_\blacksquare
\end{align}$$

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