Tenth IMAR Mathematical Competition

We begin by reviewing some basic facts on conics. For an ellipse $\Sigma$ with center $N$, foci $M$ and $M'$, semiaxes $a$ and $b$, it is known that the orthogonal projections $P$ and $P'$ of $M$ and $M'$ on any line $t$ tangent to $\Sigma$ lie on the major auxiliary circle of $\Sigma$, so that $NP = a = NP'$. Application to triangle $MNP$ (respectively, $M'NP'$) of a rotation $\theta$ (respectively, $-\theta$) about $M$ (respectively, $M'$) and a homothety of ratio $\sec\theta$ with center $M$ (respectively, $M'$) yields triangle $MOX$ (respectively, $M'OX'$), where $MO = M'O$, $NO = \frac{1}{2} \cdot MM' \cdot \tan\theta$, $OX = a \sec\theta = OX'$, and $X, X'$ both lie on $t$. If $t$ varies and $\theta$ is constant, the locus of $X$ and $X'$ is then a circle $\Gamma$ centered at $O$.

By Cartesian geometry it is readily checked that $\Gamma$ and $\Sigma$ are bitangent, and the line $\ell$ supporting their common chord is also the radical axis of the circles $\Gamma$ and $OMM'$, with this real geometrical significance even if the bitangency is not real. Since $\ell$ and the circle $OMM'$ are mutually inverse in $\Gamma$, the inverse points of $M$ and $M'$ in $\Gamma$ both lie on $\ell$. Finally, the distance $d$ between the parallel lines $\ell$ and $MM'$ is given by $d \cdot MM' = 2b^2 \tan \theta$. Similar considerations hold for a hyperbola $\Sigma$.

Consider now an isopair $M, M'$ (two isogonally conjugate in the triangle $ABC$, the foci of a conic $\Sigma$ touching its sides), and take points $X, Y, Z$ (respectively, $X', Y', Z'$) on lines $BC, CA, AB$, respectively, so that the lines $MX, MY, MZ$ (respectively, $M'X', M'Y', M'Z'$) make the same directed angle $\theta$ (respectively, $-\theta$) with the perpendiculars to $BC, CA, AB$, respectively; then the isopedal triangles $XYZ, X'Y'Z'$ of angles $\theta, -\theta$ for the isopair $M, M'$ have their Miquel points at $M, M'$ and are inscribed in a common isopedal circle $\Gamma$ bitangent to $\Sigma$, centered at a point $O$ on the perpendicular bisector of the segment $MM'$.
Conversely, for any pair of triangles inscribed in a triangle $ABC$ and in a circle $\Gamma$ (as in the statement of the problem), the Miquel points $M, M'$ are an isopair and $\Gamma$ is an isopedal circle of $M, M'$. (If $M, M'$ are the Brocard points, $\Sigma$ is the Brocard ellipse and $\Gamma$ is a Tucker circle.)