PROPOSITION 2. PROBLEM.
In a given circle, to inscribe a triangle equiangular to a given triangle.
Let ABC be the given circle, and DEF the given triangle: it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.
Draw the straight line GAH touching the circle at the point A; [III. 17. at the point A, in the straight line AH, make the angle GAB equal to the angle HAC;[I 23.
and, at the point A, in the straight line AG, make the angle GAB equal to the angle DFE; and join BC. ABC shall be the triangle required.
Because GAH touches the circle ABC, and AC is drawn from the point of contact A, [Construction.
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. [Constr.
Therefore the angle ABC is equal to the angle DEF. [Ax.l.
For the same reason the angle ACB is equal to the angle DFE.
Therefore the remaining angle BAC is equal to the remaining angle EDF. [I. 32, Axioms 11 and 3.
Wherefore the triangle ABC is equiangular to the triangle DEF, and it is inscribed in the circle ABC. q.e.f.
PROPOSITION 3. PROBLEM.
About a given circle, to describe a triangle equiangular to a given triangle.
