# Reflections of the Orthocenter II

### Vladimir Nikolin

an elementary school teacher from Serbia

April 23, 2010

Reflections of the orthocenter in the sidelines of a triangle lie on the circumcircle of the triangle.

Let H be the orthocenter in ΔABC. Let H' be the refection of H using line AB, and point D is the second end of the circumdiameter CD. M_{c} the midpoint of AB.

Since CD is a diameter, ∠CBD = 90° (as an inscribed angle subtended by a diameter), implying

Similarly, BH is an altitude (BH ⊥ AC) while ∠CAD = 90°. Therefore AD||BH.

The quadrilateral ADBH is a parallelogram. The diagonals AB and DH cross at their common midpoint which means that the midpoint M_{c} of AB lies on DH and, in addition, HM_{c} = M_{c}D.

Letting C' be the foot of the altitude CH, we have, by the construction, _{c} is the midline parallel to DH': _{c}||DH'.

**Note**: a different proof appears elsewhere.

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