How To Find The Circumcenter Of A Triangle – Step By Step Guide With Examples

circumcenter

The point of currency of the perpendicular bisectors of the sides

The circumcenter is the center of the circumcircle which is the circle that passes through all the vertices of a triangle. It is the point where the perpendicular bisectors of the sides of the triangle intersect.

To find the circumcenter of a triangle ABC, follow these steps:

1. Draw the perpendicular bisector of two sides of the triangle (e.g. AB and BC). This can be done using a straightedge and compass.

2. The point where the two perpendicular bisectors intersect is the circumcenter of the triangle.

3. Alternatively, if you have the coordinates of the three vertices of the triangle, you can use algebraic methods to find the circumcenter.

4. Let the coordinates of the vertices be A(x1, y1), B(x2, y2), and C(x3, y3).

5. Calculate the midpoints of two sides of the triangle using the midpoint formula:
Midpoint of AB = ( (x1+x2)/2 , (y1+y2)/2 )
Midpoint of BC = ( (x2+x3)/2 , (y2+y3)/2 )

6. Find the slopes of the two sides using the slope formula:
Slope of AB = (y2-y1)/(x2-x1)
Slope of BC = (y3-y2)/(x3-x2)

7. Calculate the slopes of the perpendicular lines to the two sides. The perpendicular slopes will be the negative reciprocals of the original slopes.
Slope of the line perpendicular to AB = (-1/slope of AB)
Slope of the line perpendicular to BC = (-1/slope of BC)

8. Find the equations of the two perpendicular bisectors using the point-slope form:
Equation of the perpendicular bisector of AB:
y – [(y1+y2)/2] = (-1/slope of AB)[x – [(x1+x2)/2]]

Equation of the perpendicular bisector of BC:
y – [(y2+y3)/2] = (-1/slope of BC)[x – [(x2+x3)/2]]

9. Solve the equations of the two perpendicular bisectors by setting them equal to each other and solving for x and y. The intersection point will be the circumcenter of the triangle.

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