To find the area of the equilateral triangle formed by the points (a, 1), (1, b), and (0, 0) with the constraints \(0 < a < 1\) and \(0 < b < 1\), we can follow these steps:
### Step 1: Calculate the lengths of the sides of the triangle
We will use the distance formula to find the lengths of the sides of the triangle formed by the points.
1. **Distance between (a, 1) and (1, b)**:
\[
d_1 = \sqrt{(a - 1)^2 + (1 - b)^2}
\]
2. **Distance between (1, b) and (0, 0)**:
\[
d_2 = \sqrt{(1 - 0)^2 + (b - 0)^2} = \sqrt{1 + b^2}
\]
3. **Distance between (0, 0) and (a, 1)**:
\[
d_3 = \sqrt{(a - 0)^2 + (1 - 0)^2} = \sqrt{a^2 + 1}
\]
### Step 2: Set the distances equal to each other
Since the triangle is equilateral, all sides are equal. Therefore, we can set the distances equal to each other.
1. Equate \(d_1\) and \(d_2\):
\[
\sqrt{(a - 1)^2 + (1 - b)^2} = \sqrt{1 + b^2}
\]
Squaring both sides:
\[
(a - 1)^2 + (1 - b)^2 = 1 + b^2
\]
2. Equate \(d_2\) and \(d_3\):
\[
\sqrt{1 + b^2} = \sqrt{a^2 + 1}
\]
Squaring both sides:
\[
1 + b^2 = a^2 + 1
\]
This simplifies to:
\[
b^2 = a^2
\]
Thus, \(b = a\) (since both are positive).
3. Equate \(d_1\) and \(d_3\):
\[
\sqrt{(a - 1)^2 + (1 - a)^2} = \sqrt{a^2 + 1}
\]
Squaring both sides:
\[
(a - 1)^2 + (1 - a)^2 = a^2 + 1
\]
This simplifies to:
\[
2(a - 1)^2 = a^2 + 1
\]
### Step 3: Solve for \(a\) and \(b\)
Expanding and simplifying:
\[
2(a^2 - 2a + 1) = a^2 + 1
\]
\[
2a^2 - 4a + 2 = a^2 + 1
\]
\[
a^2 - 4a + 1 = 0
\]
Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[
a = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}
\]
Given the constraints \(0 < a < 1\), we take:
\[
a = 2 - \sqrt{3}
\]
Thus, \(b = a = 2 - \sqrt{3}\).
### Step 4: Calculate the side length
Now we can calculate the length of one side of the triangle:
\[
d_2 = \sqrt{1 + b^2} = \sqrt{1 + (2 - \sqrt{3})^2}
\]
Calculating \(b^2\):
\[
b^2 = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}
\]
Thus:
\[
d_2 = \sqrt{1 + 7 - 4\sqrt{3}} = \sqrt{8 - 4\sqrt{3}}
\]
### Step 5: Calculate the area of the triangle
The area \(A\) of an equilateral triangle is given by:
\[
A = \frac{\sqrt{3}}{4} \times \text{(side length)}^2
\]
Calculating the area:
\[
A = \frac{\sqrt{3}}{4} \times (8 - 4\sqrt{3}) = \frac{\sqrt{3}}{4} \times (8 - 4\sqrt{3}) = 2\sqrt{3} - 3
\]
### Final Result
Thus, the area of the equilateral triangle formed by the points (a, 1), (1, b), and (0, 0) is:
\[
\text{Area} = 2\sqrt{3} - 3
\]
