Incentre

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This editable Main Article is under development and subject to a disclaimer.

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CC Image
Dashed lines are bisectors of the respective angles, red circle is the incircle with incentre M.

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.