In the diagram; the vertices A, B and C of triangle ABC are concyclic. EB and EC are tangents to the circle at B and C respectively. T is a point on AB such that TE is parallel to AC. BC cuts TE at F. Prove that B1 = T3. Prove that TBEC is a cyclic quadrilateral.

4.1 Prove that B_1 = T_3. Step 1: Identify angles related to the tangent-chord theorem. B_1 = A (Angle between tangent EB and chord BC is equal to the angle in the alternate segment, BAC). Step 2: Identify angles related to parallel lines. We are given that TE AC. T_3 = A (Corresponding angles, with TE AC and AB as the transversal). Step 3: Conclude the proof. Since both B_1 and T_3 are equal to A, it follows that B_1 = T_3. 4.2 Prove that TBEC is a cyclic quadrilateral. Step 1: Use the result from 4.1. From 4.1, we have proven that B_1 = T_3. Step 2: Relate these angles to the quadrilateral TBEC. In quadrilateral TBEC, B_1 is EBC. In quadrilateral TBEC, T_3 is ETC. Step 3: Apply the condition for a cyclic quadrilateral. Since EBC = ETC, these are angles subtended by the same chord EC at points B and T. If two angles subtended by the same chord (or arc) at the circumference are equal, then the four points are concyclic. Step 4: Conclude the proof. Therefore, TBEC is a cyclic quadrilateral. Drop the next question.