定比分点公式

定比分点公式是平面几何学的基本公式。若D点在B点与C点之间，向量AD即可表达成向量AB与向量AC。 A D → = | C D → | | B C → | A B → + | B D → | | B C → | A C → {\displaystyle {\overrightarrow {AD}}={\frac {|{\overrightarrow {CD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AC}}} 当A位于原点，即AD、AB、AC为位置向量，利用公式可由B点、C点得出D点的坐标。 B D → = | B D → | | B C → | B C → = | B D → | | B C → | ( A C → − A B → ) {\displaystyle {\overrightarrow {BD}}={\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {BC}}={\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}({\overrightarrow {AC}}-{\overrightarrow {AB}})} A D → = A B → + B D → = | C D → | | B C → | A B → + | B D → | | B C → | A C → {\displaystyle {\overrightarrow {AD}}={\overrightarrow {AB}}+{\overrightarrow {BD}}={\frac {|{\overrightarrow {CD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AC}}} 设 B D → = λ D C → {\displaystyle {\overrightarrow {BD}}=\lambda {\overrightarrow {DC}}} ，则 A D → = A B → + λ A C → 1 + λ {\displaystyle {\overrightarrow {AD}}={\frac {{\overrightarrow {AB}}+\lambda {\overrightarrow {AC}}}{1+\lambda }}} 考虑向量AO、AB、AE，有 A O → = | O E → | | B E → | A B → + | O B → | | B E → | A E → {\displaystyle {\overrightarrow {AO}}={\frac {|{\overrightarrow {OE}}|}{|{\overrightarrow {BE}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {OB}}|}{|{\overrightarrow {BE}}|}}{\overrightarrow {AE}}} | A O → | | A D → | A D → = | O E → | | B E → | A B → + | O B → | | B E → | ⋅ | A E → | | A C → | A C → {\displaystyle {\frac {|{\overrightarrow {AO}}|}{|{\overrightarrow {AD}}|}}{\overrightarrow {AD}}={\frac {|{\overrightarrow {OE}}|}{|{\overrightarrow {BE}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {OB}}|}{|{\overrightarrow {BE}}|}}\cdot {\frac {|{\overrightarrow {AE}}|}{|{\overrightarrow {AC}}|}}{\overrightarrow {AC}}} 考虑向量AD、AB、AC，又有 A D → = | C D → | | B C → | A B → + | B D → | | B C → | A C → {\displaystyle {\overrightarrow {AD}}={\frac {|{\overrightarrow {CD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AC}}} | A O → | | A D → | ⋅ | C D → | | B C → | = | O E → | | B E → | {\displaystyle {\frac {|{\overrightarrow {AO}}|}{|{\overrightarrow {AD}}|}}\cdot {\frac {|{\overrightarrow {CD}}|}{|{\overrightarrow {BC}}|}}={\frac {|{\overrightarrow {OE}}|}{|{\overrightarrow {BE}}|}}} | A O → | | A D → | ⋅ | B D → | | B C → | = | O B → | | B E → | ⋅ | A E → | | A C → | {\displaystyle {\frac {|{\overrightarrow {AO}}|}{|{\overrightarrow {AD}}|}}\cdot {\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}={\frac {|{\overrightarrow {OB}}|}{|{\overrightarrow {BE}}|}}\cdot {\frac {|{\overrightarrow {AE}}|}{|{\overrightarrow {AC}}|}}} | A O → | | A D → | = | O E → | | B E → | + | O B → | | B E → | ⋅ | A E → | | A C → | {\displaystyle {\frac {|{\overrightarrow {AO}}|}{|{\overrightarrow {AD}}|}}={\frac {|{\overrightarrow {OE}}|}{|{\overrightarrow {BE}}|}}+{\frac {|{\overrightarrow {OB}}|}{|{\overrightarrow {BE}}|}}\cdot {\frac {|{\overrightarrow {AE}}|}{|{\overrightarrow {AC}}|}}} 杜紫隆. 圆锥曲线问题中的“定比分点参数法”. 数学空间. 2011, (7): 第22–28页 [2014-07-17]. （原始内容存档于2014-07-26）.  楼可飞. 定比分点向量公式、向量基本定理的理解. 中学生数学. 2008, (11) [2014-03-08]. （原始内容存档于2014-03-08）.  张景中 彭翕成. 绕来绕去的向量法. 科学出版社. 2010. ISBN 9787030286741.