If the equation of the incircle of an equilateral triangle...

If the equation of the incircle of an equilateral triangle is $x^{2}+y^{2}+4 x-6 y+4=0$, then the equation of the circumcircle of the triangle is
(A) $x^{2}+y^{2}+4 x+6 y-23=0$
(B) $x^{2}+y^{2}+4 x-6 y-23=0$
(C) $x^{2}+y^{2}-4 x-6 y-23=0$
(D) none of these

Key Concepts

Relationship between Incircle and Circumcircle in an Equilateral Triangle

In an equilateral triangle, the radius of the circumcircle is exactly twice the radius of the incircle. This ratio arises from the geometry of the triangle and is pivotal in problems that require converting between the incircle and circumcircle equations.

Equilateral Triangle Geometry

An equilateral triangle has all sides equal and all angles equal, which leads to several important coincidences: the incenter, circumcenter, centroid, and orthocenter all lie at the same point. This unique property simplifies the relationship between its incircle and circumcircle.

Completing the Square

This algebraic technique is used to rewrite the equation of a circle from its general quadratic form into its standard form. By completing the square for both the x and y terms, one can easily identify the center and the radius of the circle, which is essential for analyzing and comparing different circles.

Incenter and Incircle

The incircle of a triangle is the largest circle that fits inside the triangle, tangent to all its sides. Its center, known as the incenter, is the point where the angle bisectors of the triangle intersect. Finding the equation of the incircle involves determining this center and the radius.

Circumcenter and Circumcircle

The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. Its center, the circumcenter, is found at the intersection of the perpendicular bisectors of the triangle's sides. Rewriting the circle's equation to find its geometric features is a common task in coordinate geometry.

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