A 10-point circle is associated with any general point of the ellipse. New properties of Fagnano's point (Q2474143)

In the real Euclidean plane orthogonal coordinate axes \(x\) and \(y\), \(x\cap\,y=:O\), and an ellipse \(H\) with center \(O\) and parametric description \(x=a\cos\varepsilon\), \(y=b\sin\varepsilon\), (\(a>b\), \(\varepsilon\in[0,2\pi]=:I\)) are given. Assume that \(P\in\,H\) corresponds to \(\varepsilon_P\in\,I\) and that \(P\) is distinct from the apices of \(H\). Denote by \(t\) and \(n\) the tangent resp. normal to \(H\) at \(P\); the symm-norm line \(n'\) is incident with \(P\) and has slope \(-(a/b)\tan\varepsilon_P\). The lines \(e\) and \(e'\) with \(y=x\tan\varepsilon_P\) resp. \(y=-x\tan\varepsilon_P\) are called eccentric line of \(P\) resp. symm-ecc line of \(P\). Put \(n\cap\,e=:E_1\) and \(n\cap\,e'=:I_1\) and denote the parallels to \(n'\) through \(E_1\) and \(I_1\) by \(\overline{n}\,\,'_{E_1}\) and \(\overline{n}\,\,'_{I_1}\), respectively. Finally, we need the three circles \(K_I\), \(M\) (Monge's circle), and \(K_E\) all with center \(O\) and radii \(a-b\) resp. \(\sqrt{a^2+b^2}\) resp. \(a+b\). Using only elementary facts from trigonometry and analytic geometry the author proves that the following ten points are on the same circle: 1. \(O\), 2. \(t\cap\,x\), 3. \(t\cap\,y\), 4. \(E_1\), 5. \(I_1\), 6.+7. \(M\cap\,n'\), 8. \((\overline{n}{}'_{I_1}\cap\,K_I)\setminus\{I_1\}\), 9. \((\overline{n}{}'_{E_1}\cap\,K_E)\setminus\{E_1\}\), 10. the pole of \(n'\) with regard to \(M\). If especially \(\tan\varepsilon_P=\sqrt{b/a}\), then \(P\) is Fagnano's point for which the author derives new properties. The paper contains two helpful figures.