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Base sides of trapezoid are 8 and 6. Parallel line to base sides that connects two other sides splits trapezoid to two surfaces with equal area. What is the length of this line?

My attempt: I tried drawing figure and manipulating formulas, to no avail for finding the length of this side.

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Hint

"Complete" upto a triangle and work with triangles in Thales position.

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Write area of each of the small trapezes in terms of the area of the original one. We use the bases of the original trapeze as $L_{1,2}$, the middle line is $L$, and the distance between $L$ and $L_{1,2}$ is $h_{1,2}$:$$\frac{L+L_1}{2}h_1=\frac12\frac{L_1+L_2}{2}(h_1+h_2)\\\frac{L+L_2}{2}h_2=\frac12\frac{L_1+L_2}{2}(h_1+h_2)$$ Now let's group terms with $h_1$ and $h_2$ on opposite sides of each equation: $$\begin{align}\left(\frac{L+L_1}{2}-\frac{L_1+L_2}{4}\right)h_1&=\frac{L_1+L_2}{4}h_2\\\frac{L_1+L_2}{4}h_1&=\left(\frac{L+L_2}{2}-\frac{L_1+L_2}{4}\right)h_2 \end{align}$$ Simplifying the expressions in parenthesis, and taking the ratio of the two equations yields: $$\frac{2L+L_1-L_2}{L_1+L_2}=\frac{L_1+L_2}{2L+L_2-L_1}$$ This is equivalent to $$(2L+(L_2-L_1))(2L-(L_2-L_1))=(L_1+L_2)^2$$ or: $$4L^2-(L_2-L_1)^2=(L_1+L_2)^2$$ Then the final answer is $$L=\frac12\sqrt{(L_1+L_2)^2+(L_2-L_1)^2}=\sqrt{50}$$

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