begin{equation*}
    begin{aligned}
        begin{split}
            left| left(a - cright) - frac{|b|^2cdot |d|^2}{|b times d|^2} left[
                                     left(left(a - cright)cdot left(frac{b}{|b|} - frac{b cdot d}{|b|cdot|d|} cdot frac{d}{|d|}right)right) cdot frac{b}{b} right.right. + \
            left.left. left(left(a - cright)cdot left(frac{d}{|d|} - frac{b cdot d}{|b|cdot|d|} cdot frac{b}{|b|}right)right) cdot frac{d}{d}
            right] right| = frac{|(a - c) cdot b times d|}{|b times d|}
        end{split}
    end{aligned}
end{equation*}

begin{equation*}
    begin{aligned}
        begin{split}
            left| left(a - cright) - frac{|b|^2cdot |d|^2}{|b times d|^2} left[
                                     & left(left(a - cright)cdot left(frac{b}{|b|} - frac{b cdot d}{|b|cdot|d|} cdot frac{d}{|d|}right)right) cdot frac{b}{b} right.right. + \
            & left.left. left(left(a - cright)cdot left(frac{d}{|d|} - frac{b cdot d}{|b|cdot|d|} cdot frac{b}{|b|}right)right) cdot frac{d}{d}
            right] right| = frac{|(a - c) cdot b times d|}{|b times d|}
        end{split}
    end{aligned}
end{equation*}