Let X₁, X₂, ... , X_n be n≥2 distinct points on a circle C with center O and radius r. What is the locus of points P inside C such that

   ∑ⁿ_i=1 |X_iP|/|PY_i| = n

,where the line X_iP also intersects C at point Y_i.

Let (xi,yi) be the coordinates of Xi.
Equation of C:x²+y²=r² and (a,b) be the coordinates of P.
 ∑ni=1 |XiP|/|PYi| = n 
=>  ∑ni=1 |XiP|²/|PYi|*|XiP|= n 

|PYi|*|XiP|= k(constant)
=> ∑ni=1 |XiP|² = k*n.
Consider a line through P and C(0,0). 

k=(r+PO)*(r-PO) = r²-a²-b² (Since P is inside C)

|XiP|² = (a-xi)²+(b-yi)²=a²+b²-2axi-2byi+r²
Now the eqn becomes
∑ni=1(a²+b²-2axi-2byi+r²)=(r²-a²-b²)*n
=> 2a∑ni=1 xi +2b∑ni=1 yi = 2na²+2nb²

Now, the locus of P will be
(x/n)*∑ni=1 xi +(y/n)*∑ni=1 yi = x²+y²
It is a circular arc whose center is (∑ni=1 xi/2n,∑ni=1 yi/2n)

Posted by Praneeth on 2007-12-14 08:25:37