For such a "Romain triangle", the missing theorem states that, if one of the inner angles is twice another inner angle, one has the property a² = bc +c², where a, b and c are outlined in Figure 1.
The sine chord pattern presented in the Teaching sine function with spaghetti provides helpful insights for the missing theorem. In this pattern, all angles at the intersection of chords are integer multiples of a chosen unit angle or its complement, modulo 90°. All segment lengths in this pattern can therefore be written as sums or differences of cosine and sine products and ratios of that angle. Figure 2 pictures some measures of sine chords in a circle of unit diameter, for an arbitrary angle θ.
Figure 3 shows a Romain triangle for angle α in this pattern. There are numerous other Romain triangles in this pattern. Can you figure them out? With the help of the measures pictured in Figure 2, one can follow visually the elements of a proof for the missing theorem.
Side b is the sine of the chosen angle θ (Figure 4):
b = sinθ
Side c times sin3θ equals b times sinθ (Figure 5):
c = sin²θ/sin3θ
Side a equals b cosθ minus c cos3θ (Figure 6)
a = sinθ cosθ - sin²θ cos3θ /sin3θ
Working out a² and bc +c², one finds that they are both equal to 2 sin²θ cosθ / sin3θ.
The interesting thing is that the sine chord pattern hosts plenty of "missing theorems" about angular proportions, which are only waiting to be (re)discovered.
Update (June 7, 2010): I corrected Figure 6, which held a wrong term.