MathLabs

Problem 1

Let Γ\Gamma be the circumcircle of triangle ABCABC. Let DD be a point on side BCBC. The tangent to Γ\Gamma at AA intersects the line through DD parallel to BABA at point EE. The segment CECE intersects Γ\Gamma again at FF. Suppose BB, DD, FF, EE are concyclic. Prove that ACAC, BFBF, DEDE are concurrent.
Step 1 of 4: Tangent–chord angle at A
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∠EAC=∠ABC\angle EAC=\angle ABC
Detailed analysis

Since AEAE is tangent to Γ\Gamma at AA, the tangent–chord angle theorem gives ∠EAC=∠ABC\angle EAC=\angle ABC, the inscribed angle subtending ACAC from the alternate segment.