Difference between revisions of "Menelaus' Theorem"

m (Added link to 'collinear' page.)
(Proof Using Barycentric coordinates)
Line 64: Line 64:
  
  
Which yields, after simplification,  
+
which yields, after simplification,  
  
<center> <math>-X\cdot(R-1)(P-1)+Y\cdotR(1-P)-Z\cdotPR = 0</math>
+
<cmath>-X\cdot(R-1)(P-1)+Y\cdotR(1-P)-Z\cdotPR = 0</cmath>
  
 +
<cmath>Z\cdotPR = -X\cdot(R-1)(P-1)+Y\cdotR(1-P).</cmath>
  
 +
Plugging in the coordinates for <math>Q</math> yields <math>(Q-1)(R-1)(P-1) = QPR</math>. From <math>\frac{CP}{PB}=\frac{1-P}{P},</math> we have <cmath>P=\frac{(1-P)\cdot PB}{CP}.</cmath> Likewise, <cmath>R=\frac{(1-R)\cdot AR}{BR}</cmath> and <cmath>Q=\frac{(1-Q)\cdot QC}{QA}.</cmath>
  
<math>Z\cdotPR = -X\cdot(R-1)(P-1)+Y\cdotR(1-P)</math>
 
  
 
+
Substituting these values yields <cmath>(Q-1)(R-1)(P-1) = \frac{(1-Q)\cdot QC \cdot (1-P) \cdot PB \cdot (1-R) \cdot AR}{QA\cdotCP\cdotBR}</cmath> which simplifies to <math>QA\cdot CP \cdot BR = -QC \cdot AR \cdot PB.</math>
Plugging in the coordinates for <math>Q</math> yields:
 
 
 
 
 
<math>(Q-1)(R-1)(P-1) = QPR</math>
 
 
 
 
 
We know that, since
 
 
 
 
 
<math>\frac{CP}{PB}=\frac{1-P}{P}</math>,
 
 
 
 
 
<math>P=\frac{(1-P)\cdot PB}{CP}</math>. Likewise,
 
 
 
 
 
<math>R=\frac{(1-R)\cdot AR}{BR}</math>, and
 
 
 
 
 
<math>Q=\frac{(1-Q)\cdot QC}{QA}</math>.
 
 
 
 
 
Substituting these values yields:
 
 
 
 
 
<math>(Q-1)(R-1)(P-1) = \frac{(1-Q)\cdot QC \cdot (1-P) \cdot PB \cdot (1-R) \cdot AR}{QA\cdotCP\cdotBR}</math>
 
 
 
Which simplifies to:
 
 
 
 
 
<math>QA\cdot CP \cdot BR = -QC \cdot AR \cdot PB</math>
 
  
 
QED
 
QED

Revision as of 16:48, 8 September 2014

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Menelaus' Theorem deals with the collinearity of points on each of the three sides (extended when necessary) of a triangle. It is named for Menelaus of Alexandria.

Statement

A necessary and sufficient condition for points $P, Q, R$ on the respective sides $BC, CA, AB$ (or their extensions) of a triangle $ABC$ to be collinear is that

$BP\cdot CQ\cdot AR = -PC\cdot QA\cdot RB$

where all segments in the formula are directed segments.

[asy] defaultpen(fontsize(8)); pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R; draw((0,0)--(10,0)--(7,6)--(0,0),blue+0.75); draw((7,6)--(6,8)--(4,0)); R=intersectionpoint(A--B,Q--P); dot(A^^B^^C^^P^^Q^^R); label("A",A,(1,1));label("B",B,(-1,0));label("C",C,(1,0));label("P",P,(0,-1));label("Q",Q,(1,0));label("R",R,(-1,1)); [/asy]

Proof

Draw a line parallel to $QP$ through $A$ to intersect $BC$ at $K$:

[asy] defaultpen(fontsize(8)); pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R, K=(5.5,0); draw((0,0)--(10,0)--(7,6)--(0,0),blue+0.75); draw((7,6)--(6,8)--(4,0)); draw(A--K, dashed); R=intersectionpoint(A--B,Q--P); dot(A^^B^^C^^P^^Q^^R^^K); label("A",A,(1,1));label("B",B,(-1,0));label("C",C,(1,0));label("P",P,(0,-1));label("Q",Q,(1,0));label("R",R,(-1,1)); label("K",K,(0,-1)); [/asy]

$\triangle RBP \sim \triangle ABK \implies \frac{AR}{RB}=\frac{KP}{PB}$

$\triangle QCP \sim \triangle ACK \implies \frac{QC}{QA}=\frac{PC}{PK}$

Multiplying the two equalities together to eliminate the $PK$ factor, we get:

$\frac{AR}{RB}\cdot\frac{QC}{QA}=-\frac{PC}{PB}\implies \frac{AR}{RB}\cdot\frac{QC}{QA}\cdot\frac{PB}{PC}=-1$

Proof Using Barycentric coordinates

Disclaimer: This proof is not nearly as elegant as the above one. It uses a bash-type approach, as barycentric coordinate proofs tend to be.

Suppose we give the points $P, Q, R$ the following coordinates:

$P: (0, P, 1-P)$

$R: (R , 1-R, 0)$

$Q: (1-Q ,0 , Q)$

Note that this says the following:

$\frac{CP}{PB}=\frac{1-P}{P}$

$\frac{BR}{AR}=\frac{1-R}{R}$

$\frac{QA}{QC}=\frac{1-Q}{Q}$

The line through $R$ and $P$ is given by: $\begin{vmatrix} X & 0 & R \\ Y & P & 1-R\\ Z & 1-P & 0 \end{vmatrix} = 0$


which yields, after simplification,

\[-X\cdot(R-1)(P-1)+Y\cdotR(1-P)-Z\cdotPR = 0\] (Error compiling LaTeX. Unknown error_msg)
\[Z\cdotPR = -X\cdot(R-1)(P-1)+Y\cdotR(1-P).\] (Error compiling LaTeX. Unknown error_msg)

Plugging in the coordinates for $Q$ yields $(Q-1)(R-1)(P-1) = QPR$. From $\frac{CP}{PB}=\frac{1-P}{P},$ we have \[P=\frac{(1-P)\cdot PB}{CP}.\] Likewise, \[R=\frac{(1-R)\cdot AR}{BR}\] and \[Q=\frac{(1-Q)\cdot QC}{QA}.\]


Substituting these values yields

\[(Q-1)(R-1)(P-1) = \frac{(1-Q)\cdot QC \cdot (1-P) \cdot PB \cdot (1-R) \cdot AR}{QA\cdotCP\cdotBR}\] (Error compiling LaTeX. Unknown error_msg)

which simplifies to $QA\cdot CP \cdot BR = -QC \cdot AR \cdot PB.$

QED

See also