Question 24
Let $ABC$ a triangle and $H$ his orthocentre. we note $\ell_{1}$ and $\ell_{2}$ two straight lines passing through $H$ and perpendicular to each other. we note $X_{1}, Y_{1}$ and $Z_{1}$ the points of intersection of $\ell_{1}$ with the lines $(BC)$, $(CA)$ and $(AB)$ respectively. We also note $X_{2}, Y_{2}$ and $Z_{2}$ the points of intersection of $\ell_{2}$ with the lines $(BC)$, $(CA)$ and $(AB)$ respectively. Show that the midpoints of the segments $[X_{1}X_{2}], [Y_{1}Y_{2}]$ and $[Z_{1}Z_{2}]$ are aligned.