K is the orthocenter of II_aE, excenter (Kharkiv City X 2019 Ukr)

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K is the orthocenter of II_aE, excenter (Kharkiv City X 2019 Ukr)

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#1 h Mar 22, 2020, 12:45 PM • 1 Y

Y by Mango247

In triangle $ABC$ , point $I$ is incenter , $I_a$ is the $A$ -excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$ , and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$ . Prove that $K$ is the orthocenter of triangle $II_aE$ .

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#2 h Mar 24, 2020, 3:53 AM • 1 Y

Y by Mango247

Obviously, $I_a\perp EK$ ,and as the diagram, we use Mequil’s to $\triangle EKI_a$ ,so we know that $EI,I_aK,\odot IPI_a$ are concurrent

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