The Circle

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 $C$.

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 (x_1, y_1)$accurately measures its relative mathematical position with respect to a given circle. When the general equation of a circle is written in standard form as (x, y) = x^2 + y^2 + Dx + Ey + F = 0$, the scalar power $of the fixed point$ is determined by directly substituting its coordinates into the left side of the completely expanded circle's equation.

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

The radical axis of two non-concentric circles is precisely defined as the linear locus of any given point whose tangents drawn independently to both individual circles possess equal absolute lengths. If two circles intersect, their common chord acts as the radical axis. For two equations in normalized general form $C_1(x,y)=0$ and $C_2(x,y)=0$, the equation of the radical axis is determined by algebraically subtracting the two: $C_1(x,y) - C_2(x,y) = 0$.

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

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

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.

When dealing with two distinct, non-concentric circles, the locus of a point whose tangents drawn to both circles have equal lengths is a straight line known as the radical axis. If the two circles are defined by the equations $C_1 = 0$ and $C_2 = 0$ (in general form with $x^2$ and $y^2$ coefficients normalized to 1), the equation of their radical axis is simply found by subtracting one equation from the other: $C_1 - C_2 = 0$. This subtraction eliminates the quadratic terms, leaving a linear equation. If the circles intersect, their radical axis is the straight line that passes directly through their common points of intersection.

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.