I read from here that the nine-point conic (QA-Co1) of a quadrangle $ABCD$ has eccentricity $e$ satisfying $$\frac{{{{({e^2} - 2)}^2}}}{{{e^2} - 1}} = \frac{{{{\sin }^2}(A + C)}}{{\sin A\sin B\sin C\sin D}}\tag1$$ I read from here that the Gergonne-Steiner Conic (QA-CO3) of a quadrangle $ABCD$ has eccentricity $e$ satisfying $$\frac{{{e^4}}}{{1 - {e^2}}} = \frac{{{{\sin }^2}(A + C)}}{{\sin A\sin B\sin C\sin D}}\tag2$$ The author of the posts gave a proof of (2) which is excerpt from his unpublished book. The proof uses formulas in context of Chapter 3.3 but I don't have access to it.
I hope to see proof of (1) and (2) and see why the right hand side of them are the same.