Let us prove that in both of the following cases, the three points are the vertices of an isosceles triangle:

(i) (1, 4), (4, 1) and (8, 8)

(ii) (–2, –2), (2, 2) and (4, –4)

(i) Let A → (1, 4)

B → (4, 1)

C → (8, 8)

be the vertices of a triangle.

We know that distance between two points (x₁, y₁) and (x₂, y₂) is given by .

Using the above distance formula,

AB

BC

CA

We find that BC = CA.

Since two sides of this triangle are equal, the points (7, 9), (3, –7) and (–3, 3) are the vertices of an isosceles triangle.

(ii) Let A → (-2, -2)

B → (2, 2)

C → (4, -4)

be the vertices of a triangle.

We know that distance between two points (x₁, y₁) and (x₂, y₂) is given by .

Using the above distance formula,

AB

BC

CA

We find that BC = CA.

Since two sides of this triangle are equal, the points (–2, –2), (2, 2) and (4, –4) are the vertices of an isosceles triangle.