>>16807206

s = s(θ) has three local maximums.
The second one occurs when:
3*Tan[θ/2] = 5 + 4*√7*Cos[ArcTan[3/67*√111]/3]
The depicted situation corresponds to that θ.

θ ≈ 2.75847546287800732597
s ≈ –0.5867342553508008908
t ≈ 1.6751308705666460709

vertices C, F, and G are collinear
vertices G, H, K, and L are collinear
vertices J, K, and O are collinear