(41) Intersection points between circles

[Albert Chan]

Another way is to solve for x, then get h (half chord length)

XCAS> circle1 := x^2 + y^2 - r1^2 = 0
XCAS> circle2 := (x-d)^2 + y^2 - r2^2 = 0
XCAS> chord := simplify(circle1 - circle2); // equation for chord

\((2d)x + (r_2^2 - r_1^2 - d^2)=0\\⇒ x = \large{d\over2} + {r_1^2 - r_2^2 \over 2d}\)

XCAS> p1, r1 := 13 + 4*i, 6.8
XCAS> p2, r2 := 6 + 10*i, 4.0
XCAS> d := abs(p1-p2)                                 → √85 ≈ 9.21954445729
XCAS> d1 := d/2 + (r1+r2)*(r1-r2)/(2*d)      → 6.24976648976
XCAS> h := sqrt((r1+d1)*(r1-d1))                 → 2.67963035203

Rotate/translate d1±h*i, back to intersection coordinates.

XCAS> v1 := (p2-p1)/d;   → (-7+6*i)/√85 = sign(p2-p1)
XCAS> p1 + v1*(d1+h*i)   → 6.51094321226 + 6.03276708097*i
XCAS> p1 + v1*(d1-h*i)   → 9.99870384656 + 10.1018211543*i