Parallel line in a non isosceles trapezoid cuts of equal segements

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Given the following trapezoid:

Prove that FG = HI.

The following I know: In triangle ACD: $CD/AD = FG/AG$ In triangle CDB: $CD/CB = HI/HB$ After that I'm lost

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2 Answers

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By repeated usage of the theorem of the three parallel lines and two secants: $$CD:FG=AD:AG=CB:BH=CD:IH$$

Therefore $CD:FG=CD:IH$, implying $IH\cong FG$.

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Because $$\frac{FG}{CD}=\frac{AF}{AC}=\frac{BI}{BD}=\frac{HI}{CD}.$$

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