Difference between revisions of "Gergonne's Theorem"
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| + | ==Theorem (Gergonne, 1818)== | ||
Let <math>\triangle ABC</math> be a triangle and points <math>M</math>, <math>N</math>, and <math>P</math> to be points on sides <math>BC, AC,</math> and <math>AB</math> respectively such that lines <math>AM, BN,</math> and <math>CP</math> are concurrent at point <math>O.</math> Then, <math>\frac{OM}{AM} + \frac{ON}{BN} + \frac{OP}{CP} = 1</math>. | Let <math>\triangle ABC</math> be a triangle and points <math>M</math>, <math>N</math>, and <math>P</math> to be points on sides <math>BC, AC,</math> and <math>AB</math> respectively such that lines <math>AM, BN,</math> and <math>CP</math> are concurrent at point <math>O.</math> Then, <math>\frac{OM}{AM} + \frac{ON}{BN} + \frac{OP}{CP} = 1</math>. | ||
| + | <asy> | ||
| + | size(200); | ||
| + | pair A,B,C,M,N,P,O; | ||
| + | A=(0,0); | ||
| + | B=(4,0); | ||
| + | C=(1,3); | ||
| + | M=(B+C)/2; | ||
| + | N=(A+C)/2; | ||
| + | P=(A+B)/2; | ||
| + | O=(1.5,1); | ||
| + | draw(A--B--C--cycle); | ||
| + | draw(A--M); | ||
| + | draw(B--N); | ||
| + | draw(C--P); | ||
| + | label("$A$",A,SW); | ||
| + | label("$B$",B,SE); | ||
| + | label("$C$",C,N); | ||
| + | label("$M$",M,E); | ||
| + | label("$N$",N,W); | ||
| + | label("$P$",P,S); | ||
| + | label("$O$",O,NE); | ||
| + | </asy> | ||
Latest revision as of 20:11, 9 November 2024
Theorem (Gergonne, 1818)
Let 
be a triangle and points 
, 
, and 
to be points on sides 
and 
respectively such that lines 
and 
are concurrent at point 
Then, 
.
![[asy] size(200); pair A,B,C,M,N,P,O; A=(0,0); B=(4,0); C=(1,3); M=(B+C)/2; N=(A+C)/2; P=(A+B)/2; O=(1.5,1); draw(A--B--C--cycle); draw(A--M); draw(B--N); draw(C--P); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$M$",M,E); label("$N$",N,W); label("$P$",P,S); label("$O$",O,NE); [/asy]](http://latex.artofproblemsolving.com/6/f/d/6fd59367e5b596e55251f5ecb381a5d80732c13e.png)
