MHB Geom Ch: Prove $AB=x^3$ Given $\triangle ABC$ & $\triangle AEF$

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In the given geometric configuration, triangles ABC and AEF share several key properties, including a midpoint E on segment AB and collinearity of points A, G, and F. The intersection of lines BG and EF at point C, along with the equal lengths CE, AC, AE, and FG, establishes a relationship between the segments. The problem requires proving that if AG equals x, then the length of AB must equal x cubed. This geometric proof hinges on the established conditions and relationships between the points and segments in the triangles. The conclusion is that the relationship AB = x^3 holds true under the specified conditions.
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The $\triangle ABC$ and $\triangle AEF$ are in the same plane. Between them, the following conditions hold:

1. The midpoint of $AB$ is $E$.
2. The points $A,\,G$ and $F$ are on the same line.
3. There is a point $C$ at which $BG$ and $EF$ intersect.
4. $CE=1$ and $AC=AE=FG$.

Prove that if $AG=x$, then $AB=x^3$.
 
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Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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