Let ABC be the given circle, and DEF the given triangle: it is required to describe a triangle about the circle ABC, equiangular to the triangle DEF.
Produce EF both ways to the points G, H; take K the centre of the circle ABC; [III. 1.
from K draw any radius KB;
at the point K, in the straight line KB, make the angle BKA equal to the angle DEG, and the angle BKC equal to the angle DFH; [I. 23.
and through the points A, B, C, draw the straight lines LAM, MBN, NCL, touching the circle ABC. [III. 17.
LMN shall be the triangle required.
Because LM, MN, NL touch the circle ABC at the points A, B, C, [Construction.
to which from the centre are drawn KA, KB, KC,
therefore the angles at the points A,B,C are right angles.[III.18.
And because the four angles of the quadrilateral figure AMBK are together equal to four right angles,/br>
for it can be divided into two triangles,
and that two of them KAM, KBM are right angles,
therefore the other two AKB, AMB are together equal to two right angles. [Axiom. 3.
But the angles DEG,DEF are together equal to two right angles. [I. 13.
Therefore the angles AKB, AMB are equal to the angles DEG, DEF;
of which the angle AKB is equal to the angle DEG; [Constr.
therefore the remaining angle AMB is equal to the remaining angle DEF. [Axiom 3.
In the same manner the angle LNM may be shewn to be equal to the angle DFE.
Therefore the remaining angle MLN is equal to the remaining angle EDF. [I. 32, Axioms 11 and 3.
