Problem 1

Let

\Gamma

be the circumcircle of triangle

ABC

. Let

D

be a point on side

BC

. The tangent to

\Gamma

A

intersects the line through

D

parallel to

BA

at point

E

. The segment

CE

intersects

\Gamma

again at

F

. Suppose

B

D

F

E

are concyclic. Prove that

AC

BF

DE

are concurrent.

Step 3 of 4: Introduce three circles

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\Gamma=(ABCF),\quad \omega_1=(ADCE),\quad \omega_2=(BDFE)

Detailed analysis

Besides $\Gamma$ (through $A,B,C,F$ ) and $\omega_1=(ADCE)$ from the previous step, we are given that $\omega_2=(BDFE)$ passes through $B,D,F,E$ . Each pair of these three circles shares exactly two of the marked points: $\Gamma$ and $\omega_2$ share $B,F$ ; $\Gamma$ and $\omega_1$ share $A,C$ ; $\omega_1$ and $\omega_2$ share $D,E$ .