![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
.
.
Y by
On each median of the triangle, a point is taken dividing the median in a ratio of 
, counting from the vertex. How many times is the area of the triangle with vertices at these points less than the area of the original triangle?
, counting from the vertex. How many times is the area of the triangle with vertices at these points less than the area of the original triangle?
.
-
for some 
.
so the desired point on the 
.
where 
are defined similarly as 
.![$$\frac{[P_AP_BP_C]}{[ABC]}=\begin{vmatrix}3/4&1/8&1/8\\1/8&3/4&1/8\\ 1/8&1/8&3/4\end{vmatrix}=\frac{25}{64}.$$](http://latex.artofproblemsolving.com/3/5/7/3576b0e02a525b5d18cc99ede1a0035a51a343dc.png)
