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`A̅B̅ ≅ D̅B̅ ≅ C̅B̅`

D̅E̅ ⟂ C̅B̅

A̅F̅ ⟂ C̅B̅

E̅G̅ ⟂ D̅B̅

∠ABD ≅ ∠DBC

Prove:

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`A̅F̅ ≅ E̅G̅`

I think this one is tricky, personally. There is an alternate method of solution which involves

**Spoiler:**

(I thought of putting this one in the "Mathematics" forum but this forum heading says, "For good logic/math puzzles.")

EDIT: Apparently the givens aren't showing up correctly on some displays. Thank you to Sandor for clarifying it for everyone. I like my diagrams uncolored so I'm restating the puzzle in words below.

Given:

AB = DB = CB

DE is perpendicular to CB

AF is perpendicular to CB

EG is perpendicular to DB

angle ABD is congruent to angle DBC

Prove:

AF = EG