Euler begins with the fact that, given four points A, B, C, D in the plane, they can either form a quadrilateral or a triangle with the fourth point on the interior, and that knowing which of these and knowing five of the six distances between points, the sixth distance is determined. He then uses this to find relations among the angles, such as sin(A). Euler does the same thing for cyclic quadrilaterals.
Original Source Citation
Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1782: I, pp. 3-18.
Opera Omnia Citation
Series 1, Volume 26, pp.258-269.