[i was inspired of this problem from an IMO problem,a good problem to research with ratios
]locus of point $X$
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Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
- Location:Khulna,Bangladesh.
let $ABC$ be a triangle and $X$ a point in the interior.let $AD,BE,CF$ be cevians through $X$.find all positions of $X$ such that $\frac {AX}{DX}=\frac {BX}{EX}=\frac {CX}{FX}$.
[i was inspired of this problem from an IMO problem,a good problem to research with ratios ]
[i was inspired of this problem from an IMO problem,a good problem to research with ratios
]বড় ভালবাসি তোমায়,মা
Re: locus of point $X$
The centroid is only such point.
It can be shown easily, as the three cevians divide the triangle in six parts. Then the equations between the area of the parts shows that all of them are of same area. So it must be the centroid.
It can be shown easily, as the three cevians divide the triangle in six parts. Then the equations between the area of the parts shows that all of them are of same area. So it must be the centroid.
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Re: locus of point $X$
i got it with similarity and showing these cevians median.
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