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To find circumcenter of a triangle enter values in input box by using our Circumcenter Calculator.
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Circumcenter calculator helps to calculate the circumcenter of a triangle.
How circumcenter of a triangle calculator works?
The triangle circumcenter calculator calculates the circumcenter of triangle with steps. Follow these steps to find the circumcenter using the circumcenter finder.
- Enter the coordinates for points A, B, and
- Click the Calculate button to see the result.
- Use Reset button to enter new values.
Circumcentre of a triangle calculator will instantly show you the circumcenter for the given coordinates.
What is circumcenter?
Circumcenter of a triangle is the point of intersection of all three perpendicular bisectors of the triangle. Perpendicular bisectors are nothing but the line or a ray that cuts another line segment into two equal parts at 90 degrees.
The circumcenter's position depends on the type of triangle:
- the circumcenter lies inside the triangle (acute angle)
- the circumcenter lies outside the triangle (obtuse angle)
- the circumcenter lies at the center of the hypotenuse (right angle)
How to find the circumcenter of a triangle?
Wondering how to calculate circumcenter without using circumcenter formula calculator? Well, there is no specific circumcenter formula to find it.
The circumcenter of a triangle can be found as the intersection of any two of the three perpendicular bisectors. The method to find circumcenter of triangle is given below.
- Find the midpoint of each side of the triangle.
- Find the slope of line.
- Calculate equation of line using slope and midpoint.
y-y1 = m (x-x1)
- Get the equation of all remaining lines using their midpoints and slope.
- Calculate intersection point by evaluating two bisector equations.
- The intersection point obtained in previous step is the circumcenter of triangle.
Find the circumcenter of triangle whose vertices are (5, 4), (3, 1), (6, 1).
Step 1: First of all, we will calculate the midpoint for each line of triangle. We will have lines AB, and BC.
A = (5, 4)
B = (3, 1)
C = (6, 1)
Suppose, D = midpoint of AB and E = midpoint of BC.
Midpoint of AB
D = [(x1 + x2)/2, (y1 + y2)/2]
D = [(5 + 3)/2, (4 + 1)/2]
Midpoint of AB = D = (4, 5/2)
Midpoint of BC
E = [(x1 + x2)/2, (y1 + y2)/2]
E = [(3 + 6)/2, (1 + 1)/2]
Midpoint of BC = E = (9/2, 1)
Step 2: Now, we will calculate the slope of line.
Slope of AB
Slope of AB = [(y2 - y1)/(x2 - x1)] = [(1 - 4)/(3 - 5)]
Slope of AB = 3/2
Slope of the perpendicular line to AB = -1/ Slope of AB = -1/(3/2) = -2/3
Slope of BC
Slope of BC = [(y2 - y1)/(x2 - x1)] = [(1 - 1)/(6 - 3)]
Slope of BC = 0
Slope of the perpendicular line to BC = -1/ Slope of BC = -1/0 = 0
Step 3: Find the equation of line.
Equation of the line AB
y = mx + b
Where m refers to the slope of perpendicular line. Place the value of slope of perpendicular line AB, m = -2/3 in above equation.
y = -2/3(x) + b ------------>(1)
Place the midpoint vertices of line AB, D = (4, 5/2) in equation 1.
5/2 = -2/3(4) + b
b = 5.17
Substitute b in equation 1.
y = -2/3(x) + 5.17 -------------->(2)
Equation of the line BC
y = mx + b
Substitute the value of slope of perpendicular line BC, m = 0 in above equation.
y = 0x + b
y = b ------------>(3)
Place the midpoint vertices of line BC, E = (9/2, 1) in equation 3.
b = 1
Substitute b in equation 3.
y = 1 -------------->(4)
Step 4: We have got the value of y. Now we have to find the value of x by substituting equation 4 in equation 2.
1 = -2/3(x) + 5.17
x = -3/2(1 – 5.17)
x = -3/2(-4.17)
x = 6.25
So, x = 6.25 and y = 1
Circumcenter of triangle ∆ABC = (6.25, 1)
Don’t want to waste time doing all of the above stuff? Use the circumcenter solver above to save your precious time.