Find distance between intersections of circle and line?

[Albert Chan]

Another way is to find angles of intersection.

Move the circle center to (0,0), line -> y' = 2(x' - 1) + 3 = 2x' + 1
Scale down to create a unit circle, line -> y'' = 2 x'' + 1√55

-> sin(z) = 2 cos(z) + 1√55

Use half angle formulas, t=tan(z/2), and let k=1√55:

2t/(1+t²) = 2 * (1-t²)/(1+t²) + k
2t = 2 - 2t² + k + kt²
(k-2) t² - 2t + (k+2) = 0

-> t = 0.6605, -1.7328
-> z = 66.89°, 239.98°

Distance between intersecting point = 2 r sin(Δz/2) = 2 √(55) sin(173.09°/2) ~ 14.8

Intersecting hi point = (√(55) cos(66.89°) - 1 , √(55) sin(66.89°)) ~ (1.91, 6.82)
Intersecting lo point = (√(55)cos(239.98°) - 1, √(55)sin(239.98°)) ~ (-4.71, -6.42)

Edit: this may be more accurate: |sin(Δz/2)| = |Δt| / √((1+t1²)(1+t2²))