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Shape mathematics
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(1) The cross m in D stands for MD⊥AC, and the cross b in E stands for BE⊥AC.

∵∠a=∠a,∴rtδamd∽rtδabe,

∴BD/BE=AM/AB=2/3,∴BD=2/3BE,

∫sδAMN = 1/2 * MD * AN = 1/2 * 2/3BE * 1/2AC = 1/6 * BE * AC,

sδABC = 1/2 * BE * AC,

∴sδabc=3sδamn,

∴S quadrilateral bcnm = 2s δ AMN.

∴ s δ ABC/s quadrilateral BCNM=3/2.

(2) If the diameter is AF and BF and BC are connected in G, then ∠ ABF = 90,

∴BE=√(AF^2-AB^2)=2,

∴sδabf= 1/2*ab*bf=4,

sδabf= 1/2*af*bg,∴bg=4/√5。

∴BC=8/√5,AG=√(AB^2-BG^2)=8/√5

sδABC = 1/2BC * AG = 32/√5,

∴S quadrilateral bcnm = 2/3s δ ABC = 64 √ 5/ 15.