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Difference between revisions of "Simple ratio"

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''of three points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852701.png" /> on a straight line''
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''of three points $M_1,M,M_2$ on a straight line''
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852702.png" /> such that
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The number $\lambda$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852703.png" /></td> </tr></table>
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$$\overline{M_1M}=\lambda\overline{MM_2}$$
  
One says, moreover, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852704.png" /> divides the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852705.png" /> in the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852708.png" /> are the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s0852709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s08527010.png" />, then the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s08527011.png" /> are
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One says, moreover, that $M$ divides the segment $M_1M_2$ in the ratio $\lambda$. If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of $M_1$ and $M_2$, then the coordinates of $M$ are
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085270/s08527012.png" /></td> </tr></table>
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$$x=\frac{x_1+\lambda x_2}{1+\lambda},\quad y=\frac{y_1+\lambda y_2}{1+\lambda}.$$
  
 
The simple ratio is an invariant of affine transformations.
 
The simple ratio is an invariant of affine transformations.

Latest revision as of 09:24, 13 April 2014

of three points $M_1,M,M_2$ on a straight line

The number $\lambda$ such that

$$\overline{M_1M}=\lambda\overline{MM_2}$$

One says, moreover, that $M$ divides the segment $M_1M_2$ in the ratio $\lambda$. If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of $M_1$ and $M_2$, then the coordinates of $M$ are

$$x=\frac{x_1+\lambda x_2}{1+\lambda},\quad y=\frac{y_1+\lambda y_2}{1+\lambda}.$$

The simple ratio is an invariant of affine transformations.

How to Cite This Entry:
Simple ratio. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_ratio&oldid=12601
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article