Bariz - Problem

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Tags: geometry

KereMath

09.02.2018 21:45

Let $ABCD$ be a cyclic quadrilateral and $w$ be its circumcircle. For a given point $E$ inside $w$, $DE$ intersects $AB$ at $F$ inside $w$. Let $l$ be a line passes through $E$ and tangent to circle $AEF$. Let $G$ be any point on $l$ and inside the quadrilateral $ABCD$. Show that if $\angle GAD =\angle BAE$ and $\angle GCB + \angle GAB = \angle EAD + \angle AGD + \angle ABE$ then $BC$, $AD$ and $EG$ are concurrent.

KereMath wrote: Let $ABCD$ be a cyclic quadriteral whose sides $BC$ and $AD$ are not parallel. Let $E$ be a point inside the circumcircle of $ABCD$ which is on the opposite side of the line $AD$ with respect to the point $C$. The lines $DE$ and $AB$ meet at $F$. Let $G$ be a point inside $ABCD$ and also on the line which is tangent to the circumcircle of triangle $AEF$ at $E$. If $$ \angle GAD = \angle BAE $$and $$ \angle GCB + \angle GBA = \angle EAD + \angle AGD + \angle ABE$$then show that the lines $BC$, $AD$ and $EG$ are concurrent.