*missing theorem*about angular proportions in a triangle, discovered (or rediscovered) by Leon Romain. This triangle construction is presented by user Linelites on youtube.

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

*a*² =*bc*+*c*², 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*= sin

*θ*

Side

*c*times sin*3θ*equals*b*times sin*θ*(Figure 5):*c*= sin²

*θ*/sin

*3θ*

Side

*a*equals*b*cos*θ*minus*c*cos*3θ*(Figure 6)*a*= sin

*θ*cos

*θ*- sin²

*θ*cos

*3θ*/sin

*3θ*

Working out

*a*² and*bc*+*c*², one finds that they are both equal to*2*sin²*θ*cos*θ*/ sin3*θ*.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.