Let the coordinates of A be (0, 0) and B (0, 633). C is 257 someunits (let's say meters, we don't care) from A and 390 m from B.

Distance A-C:  sqrt (x² + y²) = 257          <=> x² + y² = 257²           (1)
Distance B-C:  sqrt (x² + (633 - y)²) = 390  <=> x² + (633 - y)² = 390²   (2)
Substrating 2 from 1:  y² - (633 - y)² = 257² - 390²
<=> (y + a - y)(y - a + y) = 257² - 390²
<=> a (2 y - a) = 257² - 390²
<=> y = (257² - 390²) / 2a + a/2

Similarly:
x² + y² = 257²
<=> x = +/- sqrt ( 257² - y²)