# Incentre

In triangle geometry, the incentre of a triangle is the centre of the incircle, a circle which is within the triangle and tangent to its three sides. It is the common intersection of the three angle bisectors, which form a Cevian line system. The contact triangle has as vertices the three points of contact of the incircle with the three sides: it is the pedal triangle to the incentre. The inradius is the radius of the incircle: the area of the triangle is equal to the product of the inradius and the semi-perimeter. The incircle is tangent to the nine-point circle.

More generally, if a polygon has a single interior circle tangent to all its sides, this is the incircle of the polygon and the centre of the incircle is the incentre.

A circum quadrilateral is a quadrilateral with an inscribed circle. The condition for a quadrilateral to have an incircle is that the sums of the lengths of the pairs of opposite sides should be equal. Some content on this page may previously have appeared on Citizendium.