Inscribing in Circle Triangle Equiangular with Given
In the words of Euclid:
Let $GH$ be drawn tangent to $ABC$ at $A$.
Then $\triangle ABC$ is the given triangle.
Then from Angles made by Chord with Tangent $\angle HAC = \angle ABC$.
But $\angle HAC = \angle DEF$.
Then from Angles made by Chord with Tangent $\angle GAB = \angle ACB$.
But $\angle GAB = \angle DFE$.
So from Sum of Angles of Triangle Equals Two Right Angles it follows that the remaining angles are equal also: $\angle BAC = \angle EDF$.
Hence the result.