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定理S△ABC=21AB2·AC2-(AB·AC)2.证S△ABC=21|AB|·|AC|sin〈AB,AC〉=21|AB||AC|·1-cos2〈AB,AC〉=21|AB||AC|·1-|AABB|·|AACC|2=21|AB|2|AC|2-(AB·AC)2=21AB2·AC2-(AB·AC)2.例1已知空间三点A(1,2,3),B(2,-1,5),C(3,2,-5).试求:1)△ABC的面积;2)△ABC的AB边上的高.解1)
Theorem SΔABC=21AB2·AC2-(AB·AC)2. Proof SΔABC=21|AB|·|AC|sin=21|AB||AC|·1-|AABB|·|AACC|2=21|AB|2|AC|2-(AB·AC)2=21AB2·AC2-(AB·AC)2. Example 1 is known as three-point space A(1,2,3), B(2,-1,5), C(3,2,-5). Try to: 1) area of △ABC; 2) △ABC High on the AB side. Solution 1)