What is the height from where the two lines intersect?

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I apologize in advance for my English, but I really want to learn.

I took two lines and intersected them. |BD| and |AC| parallel but no length information

Question: Is it possible to find the length of |EF|? and how? they seem to be related but at the same time the information is missing

[BD] // [AG] Parallel

|BD| & |AG| : No length information

|BG| & |AG| & |DH| & |CH| : have length information

Question: |BG| & |AG| & |DH| & |CH| ------> |EF| related? is there an equation?

i know: |BD| and |GC| information, gives us an equation by similarity.

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