Okay, let's try to solve this problem. Hmm, we have three points: (x₁, y₁), (x₂, y₂), and (x₃, y₃). The distances between them are given as 3, 4, and 5 units. Wait, 3, 4, 5... that's a Pythagorean triplet! So maybe these points form a right-angled triangle? Let me check the distances:

The distance between x₁ and x₂ squared plus y₁ and y₂ squared is 9, which is 3². Similarly, between x₁ and x₃ is 16, which is 4², and between x₂ and x₃ is 25, which is 5². So yes, the sides are 3, 4, 5. That means the triangle is right-angled. Which angle is the right angle? Since 3² + 4² = 5², the right angle should be between the sides of length 3 and 4, so between the points (x₁, y₁) and (x₂, y₂), and (x₁, y₁) and (x₃, y₃). Wait, no, actually, the side opposite the right angle is the hypotenuse. So if the sides are 3, 4, 5, then the hypotenuse is 5. So the right angle is between the sides of 3 and 4. So the right angle is at (x₁, y₁)? Because the distances from (x₁, y₁) to the other two points are 3 and 4, and the distance between those other two points is 5. So yes, the triangle is right-angled at (x₁, y₁). That might be useful.

Now, we need to find the determinant squared of the matrix:

|x₁  y₁  1|
|x₂  y₂  1|
|x₃  y₃  1|

The determinant of a 3x3 matrix can be calculated using the rule of Sarrus or expansion by minors. But since we need the square of the determinant, maybe there's a relation involving the area of the triangle? Wait, the area of a triangle given by coordinates can be found using the determinant. The formula is (1/2)|determinant|. So if we square the determinant, it would be (2 * area)². But let's recall the exact formula.

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is given by (1/2)| determinant |. Therefore, determinant = 2 * area. So determinant squared would be (2 * area)² = 4 * area². Therefore, if we can find the area of the triangle, we can compute the desired value.

But wait, the triangle is right-angled with legs 3 and 4, so the area should be (1/2)*3*4 = 6. Then determinant squared would be 4*(6)² = 4*36 = 144. But wait, let me verify this.

Wait, hold on. The sides given are the distances between the points, but in the determinant formula, the area is based on the coordinates. However, in our case, the triangle is right-angled, so the area is indeed (1/2)*3*4 = 6. Therefore, determinant squared would be 4*36 = 144. But let me check again.

Wait, the determinant formula for area is (1/2)| determinant |. So determinant = 2 * area. Then determinant squared is (2 * area)² = 4 * area². If the area is 6, then 4*(6)^2 = 144. But hold on, is the area really 6?

Wait, the triangle is right-angled with legs 3 and 4, so the area should be (3*4)/2 = 6. So determinant squared is 4*36 = 144. So the answer should be 144. But let me think again.

Alternatively, maybe the determinant is related to the area, but perhaps there's a scaling factor involved because the coordinates might not be in a standard position. Wait, but in the determinant formula, the area is (1/2)|determinant|. So regardless of the coordinates, the area is half the absolute value of the determinant. So if we have a right-angled triangle with legs 3 and 4, then the area is 6, so determinant is 12, squared is 144. Yeah, that seems right.

But let me verify this with coordinates. Suppose we place the right-angled triangle at the origin. Let’s say (x₁, y₁) is at (0,0), (x₂, y₂) is at (3,0), and (x₃, y₃) is at (0,4). Then the determinant would be:

|0 0 1|
|3 0 1|
|0 4 1|

Calculating the determinant: 0*(0*1 - 1*4) - 0*(3*1 - 1*0) + 1*(3*4 - 0*0) = 0 - 0 + 12 = 12. So determinant is 12, squared is 144. Which matches our previous result. Therefore, the answer is 144.

But wait, in the problem, the distances are given as (x₁ - x₂)^2 + (y₁ - y₂)^2 = 9, which is 3². But if we placed (x₁, y₁) at (0,0), (x₂, y₂) at (3,0), then the distance is 3, which is correct. Similarly, (x₁, y₁) to (x₃, y₃) would be 4 if (x₃, y₃) is at (0,4). Then the distance between (x₂, y₂) and (x₃, y₃) would be 5, which is sqrt((3-0)^2 + (0-4)^2) = 5. So that works. Therefore, the determinant squared is 144. Therefore, the answer is 144.

But let me think again. The determinant formula gives the area as (1/2)|det|. So if the area is 6, then determinant is 12, squared is 144. So that's correct.

Alternatively, is there another way to compute the determinant? Let's recall that the determinant can be calculated as follows:

|x₁ y₁ 1|
|x₂ y₂ 1|
|x₃ y₃ 1|

This determinant can be expanded as:

x₁(y₂*1 - 1*y₃) - y₁(x₂*1 - 1*x₃) + 1(x₂*y₃ - x₃*y₂)

Which simplifies to:

x₁(y₂ - y₃) - y₁(x₂ - x₃) + (x₂ y₃ - x₃ y₂)

But maybe there's a better way. Alternatively, the determinant is equal to the area times 2, but with sign depending on the orientation. But since we are squaring it, the sign doesn't matter.

Alternatively, since the triangle is right-angled, we can use coordinates to compute the determinant. As we did earlier, placing the right angle at (0,0), the determinant becomes 12, so squared is 144.

Therefore, I think the answer is 144. Let me check once more.

Suppose we have three points: A(0,0), B(3,0), C(0,4). Then determinant is 12. Squared is 144. The distances between A and B is 3, A and C is 4, B and C is 5. Which matches the given conditions. So this configuration satisfies all the given equations. Therefore, the determinant squared is indeed 144.

Therefore, the answer is 144.

**Final Answer**
\boxed{144}
Therefore, the determinant squared is $\boxed{144}$.
</think>