Show that the points A(-1,2), B(5,2) & C(2,5) are the vertices of an isosceles triangle. Find the area of triangle ABC.

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