Coordinates of Intersection of two circles i am trying to find the coordinates of the intersection points of two circle.
Given value is the center coordinates and radius of both the circle
Please help without using equation substitution method.
 A: Let the center of the 2 cirlces be $O_1$ and $O_2$ respectively. Let $O_1O_2$ be distance $d$ apart. Let $A$ be a point of intersection (assuming it exists). 
Then, triangle $AO_1 O_2$ has side lengths $R_1, R_2, d$, and so we can calculate the value of $\angle AO_1O_2$ by cosine rule.
This allows us to determine the coordinates of $A$, since we know the length and the angle.

Yes, this is quite a tedious solution to work through and calculate. However, you mentioned that you didn't want equation substitution, and tagged it with trigonometry, hence this seems to be the best.
A: Let the equations of given circles be in the form:
$$ x^2 + y^2 + 2 f x + 2 g y + c = 0  \tag {1} $$
and
$$ x^2 + y^2 + 2 f_1 x + 2 g_1 y + c_1 = 0. \tag{2}$$

Subtracting you get equation of the straight line radical axis
$$ (f- f_1) x + (g- g_1) y + (c-c_1)/2 = 0 \tag{3}$$
Solution for cutting points is obtained more easily by plugging in from (3) to (1) [ or (3) to (2) ] rather than between (2) and (1) directly.
A: Try using idea from complex geometry, if you are familiar with it.
A: Find the midpoint of the centers say M. Find the slope of the line joining the 2 centers with respect to the x axis, call it S. Construct the right angle triangle joining the midpoint M, the assumed intersection, and the center of one of the circles C and then find its perpendicular sides length using Pythagoras' Theorem . Now that you have the slope S and the perpendicular length. Project the perpendicular on the Y axis and add it to the x coordinate of the center of the chosen circle C.
Please note that this solution works only for two circles having the same radii. If the radii are different, the midpoint of the two circles is not collinear with the two intersection points and the angle formed is not perpendicular.
