Poncelet Pairs via Isogonal Conjugacy

Abstract The starting point is a pair of a circle $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> and ellipse $${\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> , which is within $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> , so that there is a triangle inscribed in $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> and circumscribed about $${\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> . According to the famous Poncelet porism, we know that there is an infinite family of such triangles, so we will call $$\{D,E\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> a 3-Poncelet pair. Using isogonal conjugacy, we develop a geometric construction that produces new families of circles that also form a 3-Poncelet pair with $${\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> . In fact, any circle that is the locus of isogonal conjugates of a fixed point with respect to the family of Poncelet triangles of the original 3-Poncelet pair $$\{D,E\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> serves as an outer circle in a new 3-Poncelet pair with the same inner conic $${\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> . In the process, two special pairs of circles emerge as guide marks.