By definition, an isosceles triangle is one that has two congruent (equal) angles. To prove two angles equal on the triangle ABC, let us consider the definition of tangent:
tanx = opposite/adjacent = h/a
Noting that the y values for points A and B are equal, the height is the distance from point C to the line y = 2:
h = 5 - 2 = 3
The distance AB = 5 - -1 = 6
Let D be the point through C perpendicular to AB; i.e. the line x = 2.
AD = 2 - -1 = 3 ; DB = 5 - 2 = 3
(note that a triangle with perpendicular bisector is isosceles. But proof by definition follows)
The tangent of angle CAB = h/AD = 3/3 = 1
The tangent of angle CBA = h/DB = 3/3 = 1
Therefore the angles are congruent, and the triangle is isosceles.
The definition of the Area of a triangle is half base times height. We know both from above:
h = 3, AB = 6
0.5*h*AB = 9