Find the locus of this point P 
Find the locus of all points inside $\triangle ABC$ such that $PA^{2}+PB^{2}=PC^{2}$.
At first, i tried finding a right angled triangle and then tried to go on applying Pythagorean Theorem and finding other trivia but that didn't seem to work. I think the locus is most likely to be some line segment rather than an arc of some circle as that seems to be quite unrelated seen from an Euclidean perspective. Applying Apollonius' Theorem and Stewart's Theorem on some triangles might be the key to this problem although i could not find such triangles. I don't know whether this can be solved by trigonometry or not but since this is a problem from a chapter on euclidean geometry, i am sure that there is some clever way to look at this problem.
 A: Use vectors with $C$ at the origin. Then
$$|\mathbf{PA}|^2+|\mathbf{PB}|^2=|\mathbf{PC}|^2\implies|\mathbf P|^2-2(\mathbf A+\mathbf B)\cdot\mathbf P+|\mathbf A|^2+|\mathbf B|^2=\mathbf 0.$$
Then if $\mathbf D=\mathbf A+\mathbf B$, this rearranges to
$$|\mathbf P-\mathbf D|^2=2\mathbf A\cdot\mathbf B.$$
So as long as $\angle BCA<90^{\circ}$, the locus will be a circle with centre $D$ radius $\sqrt{2ab\cos C}$.
A: 
We extend median $CM$ such that $CM = MD$. Then please note that $MP$ is median of $\triangle CPD$ and $\triangle APB$. Applying Apollonius's theorem,
$PA^2 + PB^2 = 2(MP^2 + AM^2)$, $PC^2 + PD^2 = 2(MP^2 + CM^2)$
Subtracting second from first,
$PA^2 + PB^2 - PC^2 = 0 = PD^2 + 2 (AM^2 - CM^2)$
i.e $PD^2 + 2(\frac{c^2}{4} - \frac{1}{2}(a^2 + b^2) + \frac{c^2}{4}) = 0$
That leads to $PD = \sqrt{a^2+b^2-c^2} \ $. So the locus of point $P$ satisfying $PA^2 + PB^2 = PC^2$ is a circular arc with point $D$ being the center and radius $\sqrt{a^2+b^2-c^2}$. We also note that this is possible only if $a^2 + b^2 \geq c^2$.
A: Hint: choose a coordinate system in which $A=(0,0)$ and $B=(1,0)$. Then $C$ will have some fixed coordinates, say $C=(p,q)$. Then, if $P=(x,y)$, we have $PA^2 = x^2 + y^2$, $PB^2 = (x-1)^2 + y^2$, and so on. You’ll end up with an equation that you should be able to identify.
