Finding the Circumcenter of Triangle ABC

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is also the center of the circumscribed circle, which passes through all three vertices of the triangle. A key property of the circumcenter is that it is equidistant from each vertex of the triangle.

Let the coordinates of the vertices of triangle ABC be A(-2, 3), B(2, 1), and C(1, 2). Let the circumcenter be denoted by P(x, y).

Understanding the Circumcenter Property

Since the circumcenter P(x, y) is equidistant from A, B, and C, the distance from P to A is equal to the distance from P to B, and the distance from P to C. Mathematically, this means PA = PB = PC.

To avoid dealing with square roots, we can work with the square of the distances:

$PA^2 = PB^2$ and $PA^2 = PC^2$ (or $PB^2 = PC^2$).

Calculating Squared Distances

Using the distance formula, the square of the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $(x_2 - x_1)^2 + (y_2 - y_1)^2$. Applying this for our vertices and the circumcenter P(x, y):

• $PA^2 = (x - (-2))^2 + (y - 3)^2 = (x+2)^2 + (y-3)^2$
• $PB^2 = (x - 2)^2 + (y - 1)^2$
• $PC^2 = (x - 1)^2 + (y - 2)^2$

Setting up Equations for the Circumcenter

Now we set up equations using the equidistant property:

Equation 1: $PA^2 = PB^2$

$(x+2)^2 + (y-3)^2 = (x-2)^2 + (y-1)^2$

Expand both sides:

$(x^2 + 4x + 4) + (y^2 - 6y + 9) = (x^2 - 4x + 4) + (y^2 - 2y + 1)$

$x^2 + y^2 + 4x - 6y + 13 = x^2 + y^2 - 4x - 2y + 5$

Subtract $x^2 + y^2$ from both sides and move terms to one side:

$4x - 6y + 13 = -4x - 2y + 5$

$4x + 4x - 6y + 2y + 13 - 5 = 0$

$8x - 4y + 8 = 0$

Divide the equation by 4:

$2x - y + 2 = 0$ (Equation 1)

Equation 2: $PA^2 = PC^2$

$(x+2)^2 + (y-3)^2 = (x-1)^2 + (y-2)^2$

Expand both sides:

$(x^2 + 4x + 4) + (y^2 - 6y + 9) = (x^2 - 2x + 1) + (y^2 - 4y + 4)$

$x^2 + y^2 + 4x - 6y + 13 = x^2 + y^2 - 2x - 4y + 5$

Subtract $x^2 + y^2$ from both sides and move terms to one side:

$4x - 6y + 13 = -2x - 4y + 5$

$4x + 2x - 6y + 4y + 13 - 5 = 0$

$6x - 2y + 8 = 0$

Divide the equation by 2:

$3x - y + 4 = 0$   (Equation 2)

Solving the System of Linear Equations

We now have a system of two linear equations with two variables x and y:

• $2x - y + 2 = 0$
• $3x - y + 4 = 0$

From Equation 1, we can express y in terms of x:

$y = 2x + 2$

Substitute this expression for y into Equation 2:

$3x - (2x + 2) + 4 = 0$

$3x - 2x - 2 + 4 = 0$

$x + 2 = 0$

$x = -2$

Now substitute the value of x back into the expression for y:

$y = 2(-2) + 2$

$y = -4 + 2$

$y = -2$

Coordinates of the Circumcenter

The coordinates of the circumcenter (x, y) are (-2, -2).

Summary of Calculation Steps

Revision Table: Key Geometric Formulas

Additional Information about Triangle Centers

Besides the circumcenter, there are several other important centers in a triangle:

• Centroid: The intersection of the medians. It is the triangle's center of mass. Coordinates are the average of the vertex coordinates: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$.
• Incenter: The intersection of the angle bisectors. It is the center of the inscribed circle (incircle) which is tangent to all three sides.
• Orthocenter: The intersection of the altitudes (perpendiculars from vertices to opposite sides).

For a given triangle, the circumcenter, centroid, and orthocenter are collinear, lying on a line called the Euler line (unless the triangle is equilateral, in which case all four centers coincide).