The bisectors of the angles of a triangle are concurrent at a point that is equidistant from all three sides of the triangle, and is thus the centre of the unique circle that touches the three sides of the triangle internally. This circle is called the inscribed circle or incircle and its centre is the incentre, I.
If the lengths of the sides of the triangle are a, b, c and the radius of the incircle is r (called the inradius), then the area of the triangle ABC = area ABI + area BCI + area CAI = ra/2 + rb/2 + rc/2 = r(a + b + c)/2. In other words: area of triangle equals inradius times semiperimeter.
The incentre is centre of mass of a triangular frame formed of three uniform struts AB, BC, CA.
The points of contact of the incircle, Aa, Bb, Cc, are such that ABb = ACc, BCc = BAa, CAa = CBb since these are pairs of tangents from external points to the circle. It follows that ACc.BAa.CBb = ABb.CAa.BCc which is Ceva's condition for AAa, BBb, CCc to be concurrent. This point is called the Gergonne Point, K [It is the Lemoine point of the triangle Aa, Bb, Cc].
Denoting the lengths of the three pairs of tangents by u, v, w we have u + v = c, v + w = a, w + u = b, whence we find u = (b + c − a)/2, v = (c + a − b)/2, w = (a + b − c)/2, or in terms of semiperimeter: u = s−a, v = s−b, w = s−c.
The vertex circles centred at A, B, C, of radii u, v, w, each touches the other two and cuts the incircle at right angles at Aa, Bb, Cc. A fourth circle can be drawn touching these three. We can call it the inner circle. Its centre is not I nor K (except in the regular case). Is it on IK?.
The radii of four mutually tangent circles are related by 2(1/u² + 1/v² + 1/w²) = (1/u + 1/v + 1/w)² from which the radius of the inner circle is 1/[1/(s−a) + 1/(s−b) + 1/(s−c) + m.n.√(2D)].
A further three circles can be drawn forming with the incircle another set of four circles, each touching the other three. These intermediate circles are the vertex-circles of a triangle (the inner triangle) that has the inner circle as its incircle. (??)
Besides the incircle there are three other circles that touch all three extended sides of the triangle. These are the excircles, and their centres are called the excentres of the triangle (though they are not 'centres' of the triangle in our sense, since a regular triangle also has three excentres).
The lines joining the vertices A, B, C, to the points of contact Aa, Bb, Cc of the excircles centred at Ea, Eb, Ec with the opposite sides of the triangle are concurrent, at the Nagel Point, N.
There are also three External Nagel Points, Na, Nb, Nc where these joins meet the other vertex-to-contact lines.