C is equidistant from the points A and B.
Since A = (2a,0) and B = (0, 2b), then the coords for the point C is (a,b).
Hence distance of the point C from O is √(a^2 + b^2).
OAB is a right-angled triangle with height = 2b and length = 2a.
By Pythagoras' Theorem, the length of AB is √[(2b)^2 + (2a)^2] = 2√(a^2 + b^2)
i.e. OC is half the length of AB.
Hence C is equidistant form A, B and O.
Equation of line
For a line passing through the points (x1, y1), and (x2, y2), the equation of that straight line is given by
(y - y1)/(x - x1) = (y2 - y1)/(x2 - x1)
SInce we have the points (0,2b) and (2a,0), then the eqn of the line is,
(y - 2b)/(x - 0) = (0 - 2b)/(2a - 0)
(y - 2b)/x = -b/a
ay + bx = 2ab
Area of triangle
Area = (1/2) base times height
A = (1/2)(2a * 2b)
A = 2ab