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OK, then I'll cross post my answer.
Let's say you're trying to find the intersection points of the circles C1 and C2 where C1 has it's center point at (-9, 1) and has a radius of 7, and C2's center lies at (5, -5) and has a radius of 18.
Note that I posted the image to make clear what the variables names in the formula's are.
First calculate the distance, 'd', between the center-points of the two circles:
- d = √(|-9 - 5| + |1 - -5|)
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= 15.23
Now we calculate 'd1':
- d1 = (r1^2 - r2^2 + d^2) / 2*d
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= (7^2 - 18^2 + 15.23^2) / 2*15.23
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= (49 - 324 + 231.95) / 30.46
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= -43.05 / 30.46
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= -1.41
Now we solve 'h', which is 1/2 * 'a'
- h = √(r1^2 - a^2)
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= √(7^2 - -1.41^2)
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= √(49 - 1.99)
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= 6.86
To find point P3(x3,y3) which is the intersection of line 'd' and 'a' we use the following formula:
- x3 = x1 + (d1 * (x2 - x1)) / d
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= -9 + (-1.41 * (5 - -9)) / 15.23
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= -10.29
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y3 = y1 + (d1 * (y2 - y1)) / d
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= 1 + (-1.41 * (-5 - 1)) / 15.23
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= 1.55
Last but not least, calculate the points P4_i and P4_ii which are the intersection points of the two circles:
- x4_i = x3 + (h * (y2 - y1)) / d
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= -10.29 + (6.86 * (-5 - 1)) / 15.23
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= -12.99
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y4_i = y3 - (h * (x2 - x1)) / d
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= 1.55 - (6.86 * (5 - -9)) / 15.23
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= -4.75
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x4_ii = x3 - (h * (y2 - y1)) / d
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= -10.29 - (6.86 * (-5 - 1)) / 15.23
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= -7.59
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y4_ii = y3 + (h * (x2 - x1)) / d
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= 1.55 + (6.86 * (5 - -9)) / 15.23
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= 7.86
So, as you can see, the intersection points are (-12.99, -4.75) and (-7.59, 7.86).
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