Equal Incircles Theorem

The Equal Incircles Theorem gives conditions under which two triangles have incircles with the same radius.

Inradius: The radius of a circle inscribed in a polygon, tangent to each of its sides. Equal inradii are the theorem’s defining comparison.
Triangle: A polygon with three sides and three angles. The theorem concerns pairs of these figures.
Equilateral triangle: A triangle whose three sides and three angles are equal. Its inradius is fixed by its side length, giving a simple comparison case.
Circumcircle: The unique circle passing through all three vertices of a triangle. Its radius is not the inradius compared by this theorem.
Incircle: A circle inside a polygon that is tangent to every side. The theorem compares the radii of these circles in two triangles.
Angle bisector: A ray or line that divides an angle into two equal angles. The triangle’s angle bisectors meet at its incenter.
Isosceles triangle: A triangle with at least two equal sides. Its symmetry simplifies calculations of area and inradius.
Circumradius: The radius of a triangle’s circumcircle. Equal circumradii impose a different condition from equal inradii.
Heron’s formula: A formula giving a triangle’s area from its three side lengths. Combined with the inradius-area relation, it expresses inradius using side lengths.
Incenter: The point where a triangle’s three internal angle bisectors intersect. It is the center of the incircle whose radius the theorem compares.