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Eight times math is easy.
(1) connect BE, eab = 60, AE=AB, ∴△EAB equilateral delta, AE=AB=BE, ∫≈≈ egb = ∠ eab, ∴A,.=AG? +EG? -2AGEG/2,BE? =BG? +EG? -2beg/2, then AG? +EG? -AGEG=BG? +EG? -BGEG,AG? -BG? =AGEG-BGEG, when AG≠BG, EG=AG+BG, when AG=BG, AB⊥EG, ∠ EAG = ∠ ebg = 90, EG=2AG, EG=AG+BG holds, in a word, eg = ag+BG holds.

(2) Connect EB, ∠≈egb =∠eab, ∴A, G, B, E four-point * * circle, ∠ AEB = ∠ AGB = 45, ∠ Abe = ∠ Age. =AG? +BG? -√2AGBG,AE=AG? +EG? -√2AGEG, then AG? +BG? -√2AGBG=AG? +EG? -√2AGEG,BG? -Like what? =√2AG(BG-EG),BG & gtEG,BG+EG=√2AG .