Isosceles Triangle:
An isosceles triangle is a triangle with two sides of equal length. In this case, the base BC of the triangle has been given as B(1, 3) and C(-2, 7).

Coordinates of Vertex A:
To find the coordinates of vertex A, we need to find a point that is equidistant from both points B and C.

Midpoint of BC:
First, let's find the midpoint of BC. The midpoint formula is given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Using the given coordinates of B(1, 3) and C(-2, 7), we can calculate the midpoint as follows:
Midpoint = ((1 + (-2)) / 2, (3 + 7) / 2)
= (-1/2, 5)

Distance between Midpoint and B:
Next, let's calculate the distance between the midpoint and point B. The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using the coordinates of the midpoint (-1/2, 5) and point B(1, 3), we can calculate the distance as follows:
Distance = √((1 - (-1/2))^2 + (3 - 5)^2)
= √((3/2)^2 + (-2)^2)
= √(9/4 + 4)
= √(9/4 + 16/4)
= √(25/4)
= 5/2

Equidistant Point:
Since an isosceles triangle has two equal sides, the distance between the midpoint and point C should also be 5/2. We can calculate the distance between the midpoint and point C using the same distance formula:

Distance = √((-2 - (-1/2))^2 + (7 - 5)^2)
= √((-3/2)^2 + 2^2)
= √(9/4 + 4)
= √(9/4 + 16/4)
= √(25/4)
= 5/2

Coordinates of Vertex A:
Now that we know the midpoint of BC is equidistant from both B and C, we can find the coordinates of vertex A by reflecting the midpoint across the line BC.

To reflect a point across a line, we can use the following formula:
Reflected Point = (2 * x_of_line - x_of_point, 2 * y_of_line - y_of_point)

Using the coordinates of the midpoint (-1/2, 5) and the line BC, we can calculate the coordinates of vertex A as follows:
A = (2 * x_of_BC - x_of_midpoint, 2 * y_of_BC - y_of_midpoint)
= (2 * ((1 + (-2)) / 2) - (-1/2), 2 * ((3 + 7) / 2) - 5)
= (2 * (-1/2) - (-1/2), 2 * (