circle theorems meet 0.5absinC

an idea by Martin Wilson, in Harrogate, blending the circle theorems with area

a ppt is here

you could use normal trigonometry (by bisecting chords) but Martin's intention is that students use the (more efficient) method of calculating 0.5 a b sinC

resources 2, 3 and 4 present an interesting relationship between sines
that is not easy to justify(?)
without involving the trig. formula sinA + sinB = 2sin(semi-sum).cos(semi-difference)
and noting that cosD = sin(90 - D)

there's a sequence to follow