MathDB

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}

4