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

AI Thread Summary
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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