The standard form of the equation of a circle is derived directly from the distance formula. Let the center of the circle be the point $C(h, k)$ and the radius be $r$. Any point $P(x, y)$ on the circle must be exactly distance $r$ from $C$. Use the interactive simulation below to explore how the center $(h, k)$ and radius $r$ alter the shape and position of the circle.

If the center is precisely at the origin $(0, 0)$, the terms $h$ and $k$ become zero, and the equation simplifies dramatically to $x^2 + y^2 = r^2$.

Expanding the standard equation algebraically yields the expanded general form of the equation of a circle. By taking the standard formula $(x - h)^2 + (y - k)^2 = r^2$ and performing a full binomial expansion, we get $x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0$. By systematically grouping the resulting constant terms together, the equation simplifies into a much broader, coefficient-based structure. This specific format is particularly advantageous when dealing with systems of equations, such as determining the unique circle that passes through three distinct, non-collinear points.

To convert the general form back to the standard form and formally identify the center and radius, we use the algebraic method of completing the square for both the $x$ and $y$ groups of terms. Alternatively, you can use explicit formulas derived directly from the algebraic expansion.

A tangent to a circle is a straight line that touches the circle at exactly one point, formally known as the point of tangency.

In pure geometry, the power of a point $P(x_1, y_1)$ accurately measures its relative position with respect to a given circle. When the general equation of a circle is written as $x^2 + y^2 + Dx + Ey + F = 0$, the scalar power $p$ of the fixed point $P(x_1, y_1)$ is determined by directly substituting its coordinates into the left-hand side of the circle's equation.

If the point $P(x_1, y_1)$ lies completely outside the circle (where its power $p \gt 0$), then two distinct tangent lines can be drawn originating from $P$ to the boundary of the circle. The geometric distance $L$ from point $P$ to either of these points of tangency represents the length of the tangent. This straight-line length is mathematically equal to the square root of the power of the point.

The radical axis of two non-concentric circles is the locus of points from which tangents drawn to both circles have equal lengths. If the two circles intersect, their common chord serves as the radical axis. For two circles defined in normalized general form as $C_1(x, y) = 0$ and $C_2(x, y) = 0$ (where the coefficients of $x^2$ and $y^2$ are normalized to 1), the equation of their radical axis is found by subtracting one equation from the other: $C_1(x, y) - C_2(x, y) = 0$. This algebraic subtraction eliminates the quadratic terms, resulting in a linear equation of a straight line.

When examining three distinct, non-concentric circles whose centers do not all lie strictly on the same straight line (non-collinear centers), their three respective radical axes intersect at precisely one singular point. This intersection point is uniquely known as the radical center.

A fundamental theorem in Euclidean geometry states that any three non-collinear points define exactly one unique circle. This can be proven analytically. Since the normalized general equation $x^2 + y^2 + Dx + Ey + F = 0$ contains exactly three unknown constants ($D$, $E$, and $F$), providing the coordinates of three distinct points yields a system of three independent linear equations. Solving this system determines the parameters of the unique circle passing through all three points.

The value of the radius squared, given by the discriminant expression $r^2 = \frac{1}{4}(D^2 + E^2 - 4F)$, determines the physical nature of the resulting graph. When the equation is forced, it may not always produce a standard curve.

• Real Circle: If $r^2 \gt 0$ (meaning $D^2 + E^2 - 4F \gt 0$), the equation successfully graphs as a standard geometric circle.
• Point Circle: If $r^2 = 0$ (meaning $D^2 + E^2 - 4F = 0$), the graph collapses to a single point at $(h, k)$.
• Imaginary Circle: If $r^2 \lt 0$ (meaning $D^2 + E^2 - 4F \lt 0$), the equation has no real solutions, representing no physical points on the Cartesian plane.

Two circles are said to be orthogonal if they intersect at right angles. This implies that the tangents to the circles at their points of intersection are mutually perpendicular. For two circles given in general form $x^2 + y^2 + D_1x + E_1y + F_1 = 0$ and $x^2 + y^2 + D_2x + E_2y + F_2 = 0$, the condition for orthogonality can be derived from the Pythagorean theorem applied to the triangle formed by their centers and the point of intersection.