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If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$ and l,m,n are any positive numbers then these ratios are also equal to:
$$\left(\frac{l^2x^2 + m^2y^2 + n^2z^2}{la^2 + mb^2 + nc^2}\right)^{\frac{1}{2}}$$
$$\left(\frac{lx^2 + my^2 + nz^2}{l^2a^2 + m^2b^2 + n^2c^2}\right)^{\frac{1}{2}}$$
$$\left(\frac{lx^2 + my^2 + nz^2}{la^2 + mb^2 + nc^2}\right)^{\frac{1}{2}}$$
$$\left(\frac{lx^2 + my^2 + nz^2}{la^2 + mb^2 + nc^2}\right)$$
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