In a given circle to inscribe a triangle equiangular with a given triangle.
IN dato circulo triangulum describere dato triangulo æquiangulum.
有圜。求作圜內三角切形。與所設三角形等角。
Let ABC be the given circle, and DEF the given triangle;
thus it is required to inscribe in the circle ABC a triangle equiangular with the triangle DEF.
Let GH be drawn touching the circle ABC at A [III. 16, Por.];
on the straight line AH, and at the point A on it, let the angle HAC be constructed equal to the angle DEF,
and on the straight line AG, and at the point A on it, let the angle GAB be constructed equal to the angle DFE; [I. 23]
let BC be joined.
Then, since a straight line AH touches the circle ABC,
and from the point of contact at A the straight line AC is drawn across in the circle,
therefore the angle HAC is equal to the angle ABC in the alternate segment of the circle. [III. 32]
But the angle HAC is equal to the angle DEF;
therefore the angle ABC is also equal to the angle DEF.
For the same reason the angle ACB is also equal to the angle DFE;
therefore the remaining angle BAC is also equal to the remaining angle EDF. [I. 32]
Therefore in the given circle there has been inscribed a triangle equiangular with the given triangle.
Q. E. F.
Complete text
Book I