Olympiad Geometry Problem #91: Two Perpendiculars, Midpoint, Equal Segments

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Here is a stunning problem from the 2021 Hong Kong Team Selection Test. I found it to be surprisingly difficult, but the final solution was very elegant. I used a technique from video #63 on my channel. Enjoy! Link below.
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I still don't get why since MJF are collinear then MF=MH.

helo
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i don't understand about similar quadrilateral ?

TruongNguyen-cjpl
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I'm just curious. How long did it take you to solve this problem?

hoanglexuan