Problem 4

Triangle

ABC

has circumcircle

\Omega

and circumcenter

O

. A circle

\Gamma

with center

A

intersects the segment

BC

at points

D

and

E

, such that

B

D

E

C

are all different and lie on line

BC

in this order. Let

F

and

G

be the points of intersection of

\Gamma

and

\Omega

, such that

A

F

B

C

G

lie on

\Omega

in this order. Let

K

be the second intersection point of the circumcircle of triangle

BDF

with segment

AB

, and let

L

be the second intersection point of the circumcircle of triangle

CGE

with segment

AC

. Suppose that the lines

FK

and

GL

are distinct and intersect at the point

X

. Prove that

X

lies on the line

AO

Step 2 of 5: Introduce the second intersections with line

FG

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In plain words

Two auxiliary points on line $FG$ let us compare the two circles $(BDF)$ and $(CGE)$ through a common line.

F_2=FG\cap(BDF),\quad G_2=FG\cap(CGE)

Detailed analysis

Let line $FG$ meet circle $(BDF)$ again at $F_2$ and circle $(CGE)$ again at $G_2$ . We will relate the quadrilateral $FBDF_2$ (inscribed in $(BDF)$ ) to the quadrilateral $G_2ECG$ (inscribed in $(CGE)$ ).