Cyclic Quadrilaterals

Given two triangles ACE, BDF
intersecting at 6 points: G,H,I,J,K,L
as shown in the figure. It is given that in each of the quadrilaterals
EFGI, DELH, CDKG, BCJL, ABIK a circle can be inscribed.
Is it possible that a circle can also be inscribed in quadrilateral FAHJ?

Difficulty level (1 very easy - 10 very hard): 3

Topics:
Geometry -> Solid Geometry / Geometry in Space Geometry -> Plane Geometry -> Circles Algebra -> Equations Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation
Sources:
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