Reflections of the Orthocenter II
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. Mc the midpoint of AB.
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 Mc of AB lies on DH and, in addition, HMc = McD.
Letting C' be the foot of the altitude CH, we have, by the construction,
Note: a different proof appears elsewhere.
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