Introduction:
In this task, we need to show that the triangle formed by the points A(5,-1), B(-3,2), and C(1,6) is an isosceles triangle. To do this, we will analyze the lengths of the three sides of the triangle and compare them.

Calculating the Lengths of the Sides:
To calculate the lengths of the sides of the triangle, we will use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by √((x2-x1)² + (y2-y1)²).

Let's calculate the lengths of the three sides of the triangle:

1. Length of side AB:
- A(5,-1) and B(-3,2)
- AB = √((-3-5)² + (2-(-1))²)
- AB = √((-8)² + (3)²)
- AB = √(64 + 9)
- AB = √73

2. Length of side BC:
- B(-3,2) and C(1,6)
- BC = √((1-(-3))² + (6-2)²)
- BC = √((1+3)² + (6-2)²)
- BC = √(4² + 4²)
- BC = √(16 + 16)
- BC = √32

3. Length of side CA:
- C(1,6) and A(5,-1)
- CA = √((5-1)² + (-1-6)²)
- CA = √((5-1)² + (-7)²)
- CA = √(4² + 49)
- CA = √(16 + 49)
- CA = √65

Comparing the Lengths:
Now, let's compare the lengths of the sides AB, BC, and CA:

AB = √73
BC = √32
CA = √65

From the calculations, we can see that AB ≠ BC ≠ CA. Therefore, the lengths of the three sides of the triangle are not equal, indicating that the triangle is not equilateral.

Conclusion:
Since the triangle formed by the points A(5,-1), B(-3,2), and C(1,6) does not have all sides equal, it cannot be an equilateral triangle. However, since no two sides have the same length, it is an isosceles triangle. An isosceles triangle is a triangle with at least two sides of equal length.

first show that ab=ac not=bc