(41) Intersection points between circles

[Albert Chan]

We can also get x directly, from Law of Cosine.
Let R₂ = angle corresponded to side r₂ (the blue side)

\(r_2^2 = d^2 + r_1^2 - 2\;d\;r_1 \cos(R_2) = d^2 + r_1^2 - 2\;d\;x
\\ ⇒ x = \large{d\over2} + {r_1^2 - r_2^2 \over 2d}\)

Let Δ = area of the triangle, vertices = (0,0), (x,h), (d,0)

\(Δ = \large{d\;h\over2}\)
\(⇒ h = \large{2Δ \over d}\)