INMO 2013

In an acute angled triangle ABC with AB < AC the circle \Gamma touches AB at B and passes through C intersecting AC again at D. Prove that the orthocenter of triangle ABD lies on \Gamma if and only if it lies on the perpendicular bisector of BC.


RMO 2013 Q3Suppose H is the orthocenter of triangle ABD and it lies on the circle \Gamma . We show that HB = HC (if we can show this then the perpendicular from H on BC will bisect BC).

DF and BE are altitudes of triangle ABD.

First we note that \angle FBH = \angle HCB for FB is tangent to the circle and angle made by a chord with a tangent is equivalent to an angle in the alternate segment. In this case the chord is BH.

Again FBDE is cyclic (since \angle BFD = \angle BED = 90^0 ). Hence \angle FBH = \angle EDH (angle in the same segment FE). …. (ii)

But HDCB is also cyclic (all vertices are on the circle). Hence \angle EDH = \angle HBC (exterior angle is equal to the interior opposite angle in a cyclic quadrilateral). …. (iii)

Combining (ii) and (iii) we have \angle HCB = HBC implying HB = HC.

Conversely if we have HB = HC, this implies \angle HBC = \angle HCB . Also \angle FBD = \angle DCB (angles in the alternate segment subtended by chord BD)

Now consider triangles BEC and BFD. We have \angle BEC = \angle BFD = 90^0 and \angle ECB = \angle FBD . Therefore remaining angles BDF and EBC are also equal. But \angle DBC = \angle HCB implying \angle BDF = \angle HCB . Thus HDCB is cyclic. Hence proved.

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