Math Problem Statement

Cho đường tròn tâm O, đường kính AD. Hai dây cung AC và BD cắt nhau tại E (E nằm bên trong đường tròn (O)). Vẽ EF vuông góc với AD tại F. Chứng minh rằng: a) Tứ giác ABEF nội tiếp. b) FE là tia phân giác của BFC. c) Điểm E là tâm đường tròn nội tiếp tam giác BCF.

Solution

The image you provided seems to be a geometry problem written in Vietnamese. Here's a translation of the problem:


Question 4 (4.0 points)

Given a circle with center OO and radius ADAD. The chords ACAC and BDBD intersect at point EE, where EE lies inside the circle. The line EFEF is perpendicular to ADAD at FF. Prove the following:

a) The quadrilateral ABEFABEF is cyclic.
b) The line FEFE is the angle bisector of BFC\angle BFC.
c) Point EE is the incenter of triangle BCFBCF.


Let's break this down and start solving each part one by one. Would you like me to go through each part of the proof in detail?

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Math Problem Analysis

Mathematical Concepts

Circle Geometry
Cyclic Quadrilaterals
Angle Bisectors
Incenter

Formulas

Perpendicular line properties
Angle bisector theorem
Cyclic quadrilateral properties

Theorems

Cyclic quadrilateral theorem
Angle bisector theorem
Incenter theorem

Suitable Grade Level

Grades 10-12