向量法证明奔驰定理 .pdfVIP

向量法证明奔驰定理

英文回答：

TheInterceptTheoremisafundamentaltheoremin

geometrywhichprovidesawaytofindtheratioofthe

lengthsoftwolinesegments.Itstatesthatifa

transversalintersectstwoparallellines,thentheratio

ofthelengthsofthelinesegmentsinterceptedonone

transversalisequaltotheratioofthelengthsofthe

linesegmentsinterceptedontheothertransversal.

ToprovetheInterceptTheoremusingvectormethods,we

canusetheconceptofvectorproportionality.Vector

proportionalityisdefinedasfollows:

$$a:b=c:d\Leftrightarrow\frac{a}{b}=\frac{c}{d}

\quad\text{and}\quad\vec{a}=k\vec{b},\vec{c}=

k\vec{d}\text{forsomescalar}k$$。

Letsconsidertwoparallellines,\(l_1\)and\(l_2\),

intersectedbyatransversal\(t\).Letthepointsof

intersectionbe\(A\),\(B\),\(C\),and\(D\),asshownin

thediagrambelow:

[Imageofadiagramshowingtwoparallellines,l1and

l2,intersectedbyatransversalt.Thepointsof

intersectionareA,B,C,andD.]

Wecandefinethefollowingvectors:

$$\vec{AB}=\overrightarrow{AB},\quad\vec{BC}=

\overrightarrow{BC},\quad\vec{CD}=\overrightarrow{CD},

\quad\vec{AD}=\overrightarrow{AD}$$。

Since\(l_1\)and\(l_2\)areparallel,wehavethat

\(AD\parallelBC\).Bythedefinitionofvector

proportionality,wehave:

$$\frac{\overrightarrow{AB}}{\overrightarrow{BC}}=

\frac{\overrightarrow{AD}}{\overrightarrow{DC}}$$。

Similarly,since\(l_1\)and\(l_2\)areparallel,we

havethat\(AB\parallelCD\).Bythedefinitionofvector

proportionality,wehave:

$$\frac{\overrightarrow{AB}}{\overrightarrow{CD}}=

\frac{\overrightarrow{AD}}{\overrightarrow{BC}}$$。

Combiningthetwoequations,weget:

$$\frac{\overrightarrow{AB}}{\overrightarrow{BC}}=

\frac{\overrightarrow{AB}}{\overrightarrow{CD}}$$。

Simplifying,weget:

$$\overrightarrow{BC}=\