Difference between revisions of "Routh's Theorem"
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− | In [[triangle]] <math>ABC</math>, <math>D</math>, <math>E</math> and <math>F</math> are points on sides <math>BC</math>, <math>AC</math>, and <math>AB</math>, respectively. Let <math>r=\frac{AF}{ | + | In [[triangle]] <math>ABC</math>, <math>D</math>, <math>E</math> and <math>F</math> are points on sides <math>BC</math>, <math>AC</math>, and <math>AB</math>, respectively. Let <math>r=\frac{AF}{FB}</math>, <math>s=\frac{BD}{DC}</math>, and <math>t=\frac{CE}{AE}</math>. Let <math>G</math> be the intersection of <math>AD</math> and <math>BC</math>, <math>H</math> be the intersection of <math>BE</math> and <math>CF</math>, and <math>I</math> be the intersection of <math>CF</math> and <math>AD</math>. Then, '''Routh's Theorem''' states that |
<cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath> | <cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath> | ||
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==Proof== | ==Proof== | ||
− | {{ | + | Scale [[triangle]]<math>ABC</math>'s area down so that it becomes 1. We can then use Menelaus's Theorem on [[triangle]] <math>ABD</math> and line <math>FHC</math>. |
+ | <math>\frac{AF}{FB}\times\frac{BC}{CD}\times\frac{DG}{GA}= 1</math> | ||
+ | This means <math>\frac{DG}{GA}= \frac{BF}{FA}\times\frac{DC}{CB} = \frac{rs}{s+1}</math> | ||
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+ | Full proof: https://en.wikipedia.org/wiki/Routh%27s_theorem | ||
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+ | == See also == | ||
+ | * [[Menelaus' Theorem]] | ||
+ | *[[Ceva's Theorem]] | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
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[[Category:Theorems]] | [[Category:Theorems]] | ||
+ | [[Category:Geometry]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
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+ | {{stub}} |
Latest revision as of 23:02, 23 March 2021
In triangle , , and are points on sides , , and , respectively. Let , , and . Let be the intersection of and , be the intersection of and , and be the intersection of and . Then, Routh's Theorem states that
Proof
Scale triangle's area down so that it becomes 1. We can then use Menelaus's Theorem on triangle and line . This means
Full proof: https://en.wikipedia.org/wiki/Routh%27s_theorem
See also
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