8 If two angles in the center of a circle given, be equall at a common ray to the outter angles of a triangle given, right lines touching a periphery in the shankes of the angles, shall circumscribe a triangle about the circle given like to the triangle given. 3 p iiij.
Let there be a Triangle, and in it the outter angles aei, and aou: The Circle let it be ysr; And in the center l, let the angles ylr, and slr; at the common side lr, bee made equall to the said outter angles aei, and aou. I say the angles of the circumscribed triangle, are equall to the angles of the triangle given. For the foure inner angles of the quadrangle ylrm, are equall to the foure right angles, by the [6 e x]: And two of them, to wit, at y and r, are right angles, by the construction: For they are made by the secant and touch line, from the touch point by the center, by the [20 e xv]. Therefore the remainders at l and m, are equall to two right angles: To which two aei and aeo are equall. But the angle at l, is equall to the outter: Therefore the remainder m, is equall to aeo. The same shall be sayd of the angles aoe, and aou. Therefore two being equall, the rest at a and i, shall be equall.
Therefore
9. If a triangle be a rectangle, an obtusangle, an acute angle, the center of the circumscribed triangle is in the side, out of the sides, and within the sides: And contrariwise. 5 e iiij.
As, thou seest in these three figures, underneath, the center a.