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I am given coordinates of 3 vertices A,B and C of a triangle. Now I want to find coordinates of a point P inside this triangle if I already know PA : PB : PC.
Is there a method to determine coordinates of point P ?

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Have you find out simple way to find coordinate of P? – Nik Apr 2 '14 at 5:21
up vote 2 down vote accepted

I am afraid the co-ordinates cannot be easily determined(though, theoretically they can be if you are willing to calculate!).Here's a geometric insight into the problem.

Firstly,we shall use the angle bisector theorem and the property of the Apollonius circle.

Here in the problem,find $K'$ and $K''$ on $BC$ and $BC$ produced such that $\frac{PB}{PC}=\frac{BK'}{K'C}=\frac{BK''}{-K''C}$(we are using directed segments here).Now draw the circle $\gamma_1$ with diameter $K'K''$Now repeat the same trick with side $CA$ and draw another such circle $\gamma_{2}$. So, $\gamma_{1}\cap\gamma_{2}=P$.This suggest that you may theoretically get the co-ordinate of $P$ but I am not sure if that is a worthwhile exercise.I have no idea as to whether this will help you.

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