drawe I in the circle appointed a semidiameter, which is here H.F, for F. is the cẽtre of the circle G.H.K. Then make I on that centre an angle equall to the vtter angle B.A.D, and that angle is H.F.K. Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angle H.F.G, equall to the second vtter angle of the triangle, whiche is B.C.E. And thus haue I made .iij. semidiameters in the circle appointed. Then at the ende of eche semidiameter, I draw a touche line, whiche shall make righte angles with the semidiameter. And those .iij. touch lines mete, as you see, and make the trianagle L.M.N, whiche is the triangle that I should make, for it is drawen about a circle assigned, and hath corners equall to the corners of the triangle appointed, for the corner M. is equall to C. Likewaies L. to A, and N. to B, whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.
[ THE XXXI. CONCLVSION.]
To make a portion of a circle on any right line assigned, whiche shall conteine an angle equall to a right lined angle appointed.
The angle appointed, maie be a sharpe angle, a right angle, other a blunte angle, so that the worke must be diuersely handeled
according to the diuersities of the angles, but consideringe the hardenes of those seuerall woorkes, I wyll omitte them for a more meter time, and at this tyme wyll shewe you one light waye which serueth for all kindes of angles, and that is this. When the line is proposed, and the angle assigned, you shall ioyne that line proposed so to the other twoo lines contayninge the angle assigned, that you shall make a triangle of theym, for the easy dooinge whereof, you may enlarge or shorten as you see cause, anye of the two lynes contayninge the angle appointed. And when you haue made a triangle of those iij. lines, then accordinge to the doctrine of the seuẽ and twẽty coclusiõ, make a circle about that triangle. And so haue you wroughte the request of this conclusion. Whyche yet you maye woorke by the twenty and eight conclusion also, so that of your line appointed, you make one side of the triãgle be equal to ye ãgle assigned as youre selfe mai easily gesse.
Example.
First for example of a sharpe ãgle let A. stãd & B.C shal be ye lyne assigned. Thẽ do I make a triangle, by adding B.C, as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle is D.E.F, so yt E.F.
is the line appointed, and D. is the angle assigned. Then doo I drawe a portion of a circle about that triangle, from the one ende of that line assigned vnto the other, that is to saie, from E. a long by D. vnto F, whiche portion is euermore greatter then the halfe of the circle, by reason that the angle is a sharpe angle. But if the angle be right (as in the second exaumple you see it) then shall the portion of the circle that containeth that angle, euer more be the iuste halfe of a circle. And when the angle is a blunte angle, as the thirde exaumple dooeth propounde, then shall the portion of the circle euermore be lesse then the halfe circle. So in the seconde example, G. is the right angle assigned, and H.K. is the lyne appointed, and L.M.N. the portion of the circle aunsweryng thereto. In the third exaumple, O. is the blunte corner assigned, P.Q. is the line, and R.S.T. is the portion of the circle, that containeth that blũt corner, and is drawen on R.T. the line appointed.