case. The distance between and the orthocenter is For centroid and nine-point center we have The product of the incircle radius and the circumcircle radius of a triangle with sides is With circumradius , sides , and medians , we have If median , altitude , and internal bisector all emanate from the same vertex of a triangle with circumradius , then Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. If a triangle has two particular circles as its circumcircle and incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. (This is the case of Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality Cyclic quadrilaterals Quadrilaterals that can be circumscribed have particular properties including the fact that opposite angles are supplementary angles (adding up to 180° or π radians). Cyclic n-gons For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides are equal, and sides are equal). A cyclic pentagon with rational sides and area is known as a Robbins pentagon; in all known cases, its diagonals also have rational lengths. In any cyclic -gon with even , the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous -gon. Let one -gon be inscribed in a circle, and let another -gon be tangential to that circle at the vertices of the first -gon. Then from any point on the circle, the product of the perpendicular distances from to the sides of the first