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The problem you provided is in Hebrew and involves geometric concepts related to a circle, angles, and areas of triangles. Here's a rough translation of the question:

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**Problem 4:**

The points \( A \), \( B \), \( C \), and \( D \) lie on the circumference of a circle with radius \( R \).  
The diagonals of quadrilateral \( ABCD \) intersect at point \( E \).

Given:
- \( \angle ACD = \alpha \)
- \( \angle ABC = \alpha \)
- \( AB = AC \)
- \( BD \) bisects \( \angle ABC \)

a. The area of triangle \( \triangle ABE \) is 3 times larger than the area of triangle \( \triangle CDE \).  
   Find \( \alpha \).

b. Determine whether it is possible to inscribe a rhombus inside quadrilateral \( ABCD \).

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To proceed with solving this, let's break it down:

### Part (a):

1. You're given that the areas of triangles \( \triangle ABE \) and \( \triangle CDE \) have a specific ratio. Using this information and the properties of the circle, we can apply geometric formulas for areas and angles.
   
2. You also know that \( AB = AC \), meaning triangle \( ABC \) is isosceles. This symmetry might help in setting up the angle relationships, particularly with \( \angle ACD \) and \( \angle ABC \) both being \( \alpha \).

### Part (b):

1. This asks whether a rhombus can be inscribed within the quadrilateral. Analyzing the side lengths and angle properties of quadrilateral \( ABCD \) can help solve this.

Would you like me to work through the equations and provide the solution for \( \alpha \) and the rhombus question?

הנקודות A, B, C, ו-D נמצאות על היקף של מעגל שרדיוסו R. אלכסוני המרובע ABCD נחתכים בנקודה E. נתון: \(\angle ACD = \alpha\), \(\angle ABC = \alpha\), AB = AC, BD חוצה את הזווית \(\angle ABC\). א. שטח המשולש \(\triangle ABE\) גדול פי 3 ממשטח המשולש \(\triangle CDE\). מצאו את \(\alpha\). ב. קבעו אם ניתן לחרוס מעגל במרובע ABCD.

Solve a complex geometric problem involving points on a circle, angle relationships, and area ratios of triangles. Determine if a rhombus can be inscribed in the quadrilateral. Suitable for Grades 10-12.

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