AoPS - Problem

Source: Nordic Mathematical Contest 2013 #4

Tags: geometry, geometric transformation, reflection, Concyclic

parmenides51

23.09.2017 16:01

Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and ${H_{c}^{'} }$ are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.

If at least two of them coincide, then obviously they are concyclic. So, let's assssume that no two of them coincide. Let $HH_c \cap AB = G$, $HH_b \cap AC = F$ and $AH \cap BC = P$. Then a homothety with factor $\dfrac {1} {2}$ sends $H_b, H_c, H'_c, H'_b$ to $F, G, D, E$, respectively. So, $H_b, H_c, H'_c, H'_b$ are concyclic $\Leftrightarrow$ $F, G, D, E$ are concyclic $\Leftrightarrow$ $\measuredangle AHG = \measuredangle AFG = \measuredangle EDA = \measuredangle CBA = \measuredangle PBG$ $\Leftrightarrow$ $B, G, H, P$ are concyclic $\Leftrightarrow$ $AH \perp BC$.