# A triangular region is bound by the three lines 2x+y-12=0, y=3x-3, and y=1/7x-81/7. What is the circumcentre of this triangle?

I can find the answer using a graph, but I don't know the formula to find the mid segments of the lines.

## What you'll need to do is

1. find the 3 intersections of the lines, which are then the vertices, (xa,ya), (xb,yb), (xc,yc).
2. Find the mid-points of each side using
Mab=((xa+xb)/2, (ya+yb)/2),
...
3. Find the perpendicular bisector of each side by passing a line perpendicular to the side, and passing through the mid point.
4. The Circumcentre is the intersection of any two of these perpendicular bisectors.

For a numerical example, see:
http://www.algebra.com/algebra/homework/Length-and-distance.faq.question.524163.html

## To find the circumcentre of a triangle, you need to find the intersection point of the perpendicular bisectors of its three sides.

To begin, let's find the equations of the three lines:
1. The first line is given as 2x + y - 12 = 0. We can rewrite this equation in the form y = mx + c by isolating y, which gives y = -2x + 12.
2. The second line is y = 3x - 3.
3. The third line is y = (1/7)x - (81/7).

Now, let's find the midpoints of the sides of the triangle by finding the average of their respective coordinates:

For the first side:
Midpoint = (x1 + x2) / 2 , (y1 + y2) / 2
where (x1, y1) and (x2, y2) are the coordinates of any two points on the line.

Using the first two points on the first line (2x + y - 12 = 0), we have:
(x1, y1) = (0, 12)
(x2, y2) = (6, 0)

Midpoint of the first side = (0 + 6) / 2 , (12 + 0) / 2 = 3, 6

Similarly, find the midpoints of the other two sides using the second and third lines.

Now, we have the coordinates of the midpoints of the three sides.

The next step is to find the equations of the perpendicular bisectors of the sides. To do this, we will find the negative reciprocal of the slopes of the sides and plug in the coordinates of the midpoints:

For the first side, the slope is -2. Thus, the slope of the perpendicular bisector will be 1/2.
Using the midpoint (3, 6), we can write the equation of the perpendicular bisector of the first side as y = (1/2)x + b by substituting the values of x and y from the midpoint into the equation. Then, solve for b.

Similarly, find the equations of the perpendicular bisectors of the other two sides using their respective midpoints.

Once you have the equations of the three perpendicular bisectors, solve the system of equations to find the intersection point. This intersection point will be the circumcentre of the triangle.

Note: Solving the system of equations might require using substitution or elimination methods, depending on the equations.

Once you find the coordinates of the circumcentre, you will have your answer.