Finding locus of centroid

Question

Let AB be a chord of circle x^2 + y^2 = 3 which subtends 45 angle at P where P is any moving point on the circle. Then find the locus of centroid of triangle PAB

Any help would be appreciated

Answer 1

Take the points A, B to be $(-\sqrt\frac{3}{2},\sqrt\frac{3}{2})$ and $(-\sqrt\frac{3}{2},-\sqrt\frac{3}{2})$ let the variable point P be given by: $ ( \sqrt 3 \cos t,\sqrt 3 \sin t) $ for $ t \in [0,2\pi) $

The locus of the centroid is the circle in red. Equation: $(x + \frac{2}{3}\sqrt\frac{3}{2} )^2 + y^2 = \frac{1}{3} $

(The centroid is the sum of the coordinate components of the three points divided by 3.)