To solve the problem step by step, we will follow the logical reasoning as described in the video transcript.
### Step 1: Understand the Geometry of the Ellipse
We start with an ellipse centered at the origin (0, 0) with the major axis along the x-axis and the minor axis along the y-axis. The coordinates of the foci are given by (ae, 0) and (-ae, 0), where 'a' is the semi-major axis, 'b' is the semi-minor axis, and 'e' is the eccentricity.
### Step 2: Define the Ellipse Equation
The standard equation of the ellipse is given by:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
where \(a > b\).
### Step 3: Identify the Points
The extremities of the minor axis are at (0, b) and (0, -b). The foci are at (ae, 0) and (-ae, 0).
### Step 4: Form the Equilateral Triangle
The points A (ae, 0), B (0, b), and C (0, -b) form an equilateral triangle. Since all sides of an equilateral triangle are equal, we can set the distances between these points equal.
### Step 5: Calculate the Lengths of the Sides
1. Distance AB:
\[
AB = \sqrt{(ae - 0)^2 + (0 - b)^2} = \sqrt{a^2e^2 + b^2}
\]
2. Distance AC:
\[
AC = \sqrt{(ae - 0)^2 + (0 + b)^2} = \sqrt{a^2e^2 + b^2}
\]
3. Distance BC:
\[
BC = \sqrt{(0 - 0)^2 + (b - (-b))^2} = \sqrt{(2b)^2} = 2b
\]
### Step 6: Set the Distances Equal
Since AB = AC = BC, we have:
\[
\sqrt{a^2e^2 + b^2} = 2b
\]
### Step 7: Square Both Sides
Squaring both sides gives:
\[
a^2e^2 + b^2 = 4b^2
\]
This simplifies to:
\[
a^2e^2 = 3b^2
\]
### Step 8: Express b^2 in terms of a^2 and e^2
From the equation \(a^2e^2 = 3b^2\), we can express \(b^2\) as:
\[
b^2 = \frac{a^2e^2}{3}
\]
### Step 9: Use the Eccentricity Formula
The formula for the eccentricity of an ellipse is:
\[
e = \sqrt{1 - \frac{b^2}{a^2}}
\]
Substituting \(b^2\) from the previous step:
\[
e = \sqrt{1 - \frac{\frac{a^2e^2}{3}}{a^2}} = \sqrt{1 - \frac{e^2}{3}}
\]
### Step 10: Solve for e
Squaring both sides gives:
\[
e^2 = 1 - \frac{e^2}{3}
\]
Rearranging this gives:
\[
e^2 + \frac{e^2}{3} = 1
\]
Multiplying through by 3:
\[
3e^2 + e^2 = 3 \implies 4e^2 = 3 \implies e^2 = \frac{3}{4}
\]
Thus, the eccentricity \(e\) is:
\[
e = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}
\]
### Final Answer
The eccentricity of the ellipse is:
\[
\boxed{\frac{\sqrt{3}}{2}}
\]
