Circumcircle of a Triangle: Definitions and Examples

Circumcircle of a Triangle: Definitions and Examples Definitions, Formulas, & Examples

GET TUTORING NEAR ME!

(800) 434-2582

By submitting the following form, you agree to Club Z!'s Terms of Use and Privacy Policy

    Home / Educational Resources / Math Resources / Circumcircle of a Triangle: Definitions and Examples

    Circumcircle of a Triangle

    A circumcircle is a circle that passes through all three vertices of a triangle. In other words, it is the circle that is inscribed within the triangle and has its center on the line that is equidistant from all three vertices of the triangle.

    The circumcircle of a triangle can be constructed using a simple process. First, find the midpoint of each side of the triangle. The midpoints will form a smaller triangle within the original triangle. The center of the circumcircle is at the intersection of the perpendicular bisectors of the sides of this smaller triangle.

    To find the radius of the circumcircle, draw a perpendicular from each vertex of the triangle to the center of the circumcircle. The length of each of these perpendiculars is equal to the radius of the circumcircle.

    The circumcenter of a triangle has several interesting properties. For example, it is the center of symmetry of the triangle, which means that any line drawn from the center to a vertex is perpendicular to the opposite side of the triangle. Additionally, the circumcenter of a triangle is equidistant from all three vertices, so it is the center of the circle that is circumscribed about the triangle.

    Another important property of the circumcenter is that it is the center of the incircle of the triangle, which is the largest circle that can be inscribed within the triangle. The incircle is tangent to all three sides of the triangle at the points where they are closest to the center of the circle.

    In terms of applications, the circumcenter of a triangle is used in computer graphics to calculate the orientation of a triangle in three-dimensional space. It is also used in geometric algorithms to solve problems related to triangle circumscribing and in finding the centers of triangles and other geometric shapes.

    In conclusion, the circumcenter of a triangle is a fundamental concept in geometry with a wide range of applications in various fields, including computer graphics, mathematics, and engineering. Understanding the properties and applications of the circumcenter is essential for students and professionals who work with triangles and other geometric shapes.

    Definitions:

    Triangle: A triangle is a 2-dimensional shape made up of three straight lines and three angles.

    Circumcenter: The circumcenter is the center of a circumcircle.

    Circumcircle: A circumcircle is a circle that passes through all the vertices of a geometric shape.

    Circumscribed: A shape is said to be circumscribed if it is surrounded by a circle.

    Vertex: A vertex is a point where two or more lines meet.

    The circumcenter of a triangle is the point at which the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from all three vertices of the triangle. The radius of the circumcircle is equal to the distance between the circumcenter and any of the vertices of the triangle.

    Examples:

    1. Consider an equilateral triangle with sides of length 10 units. The circumcenter of this triangle will be located at the midpoint of each side and the radius of the circumcircle will be 5 units.
    2. Consider a right-angled triangle with sides of length 6 units, 8 units and 10 units. The circumcenter of this triangle will be located at the midpoint of the hypotenuse and the radius of the circumcircle will be 4 units.
    3. Consider an isosceles triangle with sides of length 10 units and 8 units. The circumcenter of this triangle will be located at the midpoint of the two equal sides and the radius of the circumcircle will be 4 units.
    4. Consider a scalene triangle with sides of length 5 units, 7 units and 9 units. The circumcenter of this triangle will be located at the intersection of the perpendicular bisectors of the sides of the triangle and the radius of the circumcircle will be 4 units.
    5. Consider an isosceles right-angled triangle with sides of length 6 units and 6 units. The circumcenter of this triangle will be located at the midpoint of the hypotenuse and the radius of the circumcircle will be 3 units.

    Quiz:

    1. What is a triangle?
    2. What is a circumcenter?
    3. What is a circumcircle?
    4. What does circumscribed mean?
    5. What is a vertex?
    6. Where is the circumcenter of a triangle located?

    Answers:

    1. A triangle is a 2-dimensional shape made up of three straight lines and three angles.
    2. The circumcenter is the center of a circumcircle.
    3. A circumcircle is a circle that passes through all the vertices of a geometric shape.
    4. A shape is said to be circumscribed if it is surrounded by a circle.
    5. A vertex is a point where two or more lines meet.
    6. The circumcenter of a triangle is the point at which the perpendicular bisectors of the sides of the triangle intersect.

    If you’re interested in online or in-person tutoring on this subject, please contact us and we would be happy to assist!

    Circumcircle of a Triangle:

    Result

    the circle centered at {c/2, (c (a^2 + b^2 - c^2))/(2 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)))} with radius (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    Definition

    Defining inequalities

    y>=0 and y (a^2 + c^2) + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))<=c sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) + b^2 y and a^2 y + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))>=y (b^2 + c^2)

    Lamina properties

    (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)

    3

    a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

    (data not available)

    sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

    A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))

    Mechanical properties

    J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

    J_y invisible comma y = (sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)

    J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

    J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)

    r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))

    Distance properties

    a | b | c

    p = a + b + c

    r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

    R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    max(a, b, c)

    χ = 1

    s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)

    A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

    Find the right fit or it’s free.

    We guarantee you’ll find the right tutor, or we’ll cover the first hour of your lesson.