180t = 360p + 180qor simplifying
t = 2p + qNow count the total number of sides of the triangles. Each interior side is on two triangles whereas those on the circle are only on one triangle.
3t = 2e + qSubtracting the previous equation t=2p+q from this gives;
2t = 2e - 2pWhich is the same as
t + p = eHowever by Euler's formula F+E=V+2 therefore
t+1 + p+q = e+q + 2which gives
t + p = e + 1and subtracting t+p=e from this gives the classical contradiction
0 = 1QED we can't divide up a circle into right wiggly triangles, it is just plain impossible.
A single right wiggly triangle is sufficient. The sides AB and BC are bent back on themselves so corners A, B and C all meet at a point. AC forms the outside of the circle. Angles A+B+C make up 180°.