To drawe a circle within a triangle appoincted.
For this conclusion and all other lyke, you muste vnderstande,
that when one figure is named to be within an other, that it is
not other waies to be vnderstande, but that eyther euery syde of
the inner figure dooeth touche euerie corner of the other, other
els euery corner of the one dooeth touche euerie side of the
other. So I call that triangle drawen in a circle, whose corners
do touche the circumference of the circle. And that circle is
contained in a triangle, whose circumference doeth touche
iustely euery side of the triangle, and yet dooeth not crosse
ouer any side of it. And so that quadrate is called properly to
be drawen in a circle, when all his fower angles doeth touche
the edge of the circle, And that circle is drawen in a quadrate,
whose circumference doeth touche euery side of the quadrate, and
lykewaies of other figures.
_Examples are these. A.B.C.D.E.F._
[Illustration:
A. is a circle in a triangle.
B. a triangle in a circle.
C. a quadrate in a circle.
D. a circle in a quadrate.]
In these .ij. last figures E. and F, the circle is not named to
be drawen in a triangle, because it doth not touche the sides of
the triangle, neither is the triangle counted to be drawen in the
circle, because one of his corners doth not touche the
circumference of the circle, yet (as you see) the circle is
within the triangle, and the triangle within the circle, but
nother of them is properly named to be in the other. Now to come
to the conclusion. If the triangle haue all .iij. sides lyke,
then shall you take the middle of euery side, and from the
contrary corner drawe a right line vnto that poynte, and where
those lines do crosse one an other, there is the centre. Then
set one foote of the compas in the centre and stretche out the
other to the middle pricke of any of the sides, and so drawe a
compas, whiche shall touche euery side of the triangle, but
shall not passe with out any of them.
_Example._
The triangle is A.B.C, whose sides I do part into .ij. equall
partes, eche by it selfe in these pointes D.E.F, puttyng F.
betwene A.B, and D. betwene B.C, and E. betwene A.C. Then draw I
a line from C. to F, and an other from A. to D, and the third
from B. to E.
[Illustration]
And where all those lines do mete (that is to saie M. G,) I set
the one foote of my compasse, because it is the common centre,
and so drawe a circle accordyng to the distaunce of any of the
sides of the triangle. And then find I that circle to agree
iustely to all the sides of the triangle, so that the circle is
iustely made in the triangle, as the conclusion did purporte.
And this is euer true, when the triangle hath all thre sides
equall, other at the least .ij. sides lyke long. But in the
other kindes of triangles you must deuide euery angle in the
middle, as the third conclusion teaches you. And so drawe lines
from eche angle to their middle pricke. And where those lines do
crosse, there is the common centre, from which you shall draw a
perpendicular to one of the sides. Then sette one foote of the
compas in that centre, and stretche the other foote accordyng to
the length of the perpendicular, and so drawe your circle.
[Illustration]
_Example._
The triangle is A.B.C, whose corners I haue diuided in the
middle with D.E.F, and haue drawen the lines of diuision
A.D. B.E, and C.F, which crosse in G, therfore shall G. be the
common centre. Then make I one perpendicular from G. vnto the
side B.C, and that is G.H. Now sette I one fote of the compas in
G, and extend the other foote vnto H. and so drawe a compas,
whiche wyll iustly answere to that triangle according to the
meaning of the conclusion.