The vertices of triangle abc are a(1,1), b(0,2), and c(3,-2). find the coordinates of the circumcenter.

To find the circumcenter, solve any two bisector equations and find out the intersection points. The given are A(1,1), B(0,2), and C(3,-2).

Midpoint of AB = (1/2, 3/2)  - You can get the midpoint by getting the average of x-coordinates and y-coordinates. 

Slope of AB = -1
Slope of perpendicular bisector = 1
Equation of AB with slope 1 and the coordinates (1/2, 3/2) is
y - (3/2) = (1)(x - 1/2) 
y = x+1

Do the same for AC
Midpoint of AC = (2, -1/2)
Slope of AC = -3/2
Slope of perpendicular bisector = 2/3
Equation of AC with slope 2/3 and the coordinates (2, -1/2) is
y - (-1/2) = (2/3)(x - 2) 
y = -11/6 + 2x/3

So the perpendicular bisectors of AB and BC meet
y = x+1
y = -11/6 + 2x/3

To solve for x,
(-11/6 + 2x/3) = (x+1)
x= -17/2

Now get y by substituting 
y = (-17/2) + 1
y = -15/2

The circumcenter is (-17/2, -15/2)

Thank you for posting your question. I hope that this answer helped you. Let me know if you need more help. 

Using the geometric mean concept, it is found that the value of a is 18.

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The geometric mean, of a data-set of n elements, , is given by:

That is, the nth root of the multiplication of all elements.

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In this question:

Two elements(n = 2), a and 34.

Thus:

We find the square of each side, so:

Simplifying both sides by 17:

The value of a is 18.

A similar example is given at