Solution of Triangles: Circles connected with a triangle - circumcircle

The study of triangles and the relationships between their sides and angles is a fundamental aspect of trigonometry. One important concept in this area is the circumcircle of a triangle, which is the unique circle that passes through all three vertices of the triangle. This circle is also known as the circumscribed circle.

Circumcircle of a Triangle

The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. The center of this circle is called the circumcenter, and its radius is known as the circumradius.

Properties of the Circumcircle

• The circumcenter is equidistant from all three vertices of the triangle.
• The circumcenter can lie inside, outside, or on the triangle, depending on whether the triangle is acute, obtuse, or right-angled, respectively.
• The circumradius can be found using the formula involving the sides of the triangle and the area of the triangle.

Formulas Related to Circumcircle

The circumradius $R$ of a triangle with sides $a$, $b$, and $c$ and area $\Delta$ can be calculated using the formula:

$$ R = \frac{abc}{4\Delta} $$

Alternatively, if the angles of the triangle are known, the circumradius can be found using the Law of Sines:

$$ R = \frac{a}{2\sin(A)} = \frac{b}{2\sin(B)} = \frac{c}{2\sin(C)} $$

where $A$, $B$, and $C$ are the angles opposite sides $a$, $b$, and $c$, respectively.

Circumcenter Coordinates

The coordinates of the circumcenter $(X, Y)$ for a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be found using the following determinants:

$$ X = \frac{D_x}{D} \quad \text{and} \quad Y = \frac{D_y}{D} $$

where

$$ D = \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \ \end{vmatrix}, \quad D_x = \begin{vmatrix} x_1^2 + y_1^2 & y_1 & 1 \ x_2^2 + y_2^2 & y_2 & 1 \ x_3^2 + y_3^2 & y_3 & 1 \ \end{vmatrix}, \quad D_y = \begin{vmatrix} x_1^2 + y_1^2 & x_1 & 1 \ x_2^2 + y_2^2 & x_2 & 1 \ x_3^2 + y_3^2 & x_3 & 1 \ \end{vmatrix} $$

Examples

Example 1: Finding the Circumradius

Given a triangle with sides $a = 7$, $b = 24$, and $c = 25$, find the circumradius.

First, we calculate the semi-perimeter $s$:

$$ s = \frac{a + b + c}{2} = \frac{7 + 24 + 25}{2} = 28 $$

Next, we find the area $\Delta$ using Heron's formula:

$$ \Delta = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{28(28 - 7)(28 - 24)(28 - 25)} = 84 $$

Now, we can find the circumradius $R$:

$$ R = \frac{abc}{4\Delta} = \frac{7 \cdot 24 \cdot 25}{4 \cdot 84} = \frac{4200}{336} = 12.5 $$

Example 2: Circumcenter Coordinates

Consider a triangle with vertices at $(0, 0)$, $(6, 0)$, and $(0, 8)$. Find the coordinates of the circumcenter.

Using the determinants for $D$, $D_x$, and $D_y$:

$$ D = \begin{vmatrix} 0 & 0 & 1 \ 6 & 0 & 1 \ 0 & 8 & 1 \ \end{vmatrix} = 0(0 \cdot 1 - 1 \cdot 8) - 0(6 \cdot 1 - 1 \cdot 0) + 1(6 \cdot 8 - 0 \cdot 0) = 48 $$

$$ D_x = \begin{vmatrix} 0 & 0 & 1 \ 36 & 0 & 1 \ 0 & 8 & 1 \ \end{vmatrix} = 0(0 \cdot 1 - 1 \cdot 8) - 0(36 \cdot 1 - 1 \cdot 0) + 1(0 \cdot 8 - 36 \cdot 0) = 0 $$

$$ D_y = \begin{vmatrix} 0 & 0 & 1 \ 36 & 6 & 1 \ 0 & 0 & 1 \ \end{vmatrix} = 0(6 \cdot 1 - 1 \cdot 0) - 0(36 \cdot 1 - 1 \cdot 0) + 1(0 \cdot 0 - 36 \cdot 6) = -216 $$

Therefore, the coordinates of the circumcenter are:

$$ X = \frac{D_x}{D} = \frac{0}{48} = 0 \quad \text{and} \quad Y = \frac{D_y}{D} = \frac{-216}{48} = -4.5 $$

So, the circumcenter is at $(0, -4.5)$.

Comparison Table

Property	Acute Triangle	Right Triangle	Obtuse Triangle
Circumcenter Location	Inside the triangle	On the hypotenuse	Outside the triangle
Circumradius	Smaller than any side	Half the length of the hypotenuse	Larger than the longest side
Calculation Method	Use sides and area or Law of Sines	Use the hypotenuse directly	Use sides and area or Law of Sines

Understanding the circumcircle and its properties is essential for solving problems related to triangles, especially in geometry and trigonometry.