Bisectrices and incentre

The three bisector lines of the inner angles of a triangle (the bisectrices) intersect in a unique point, the incentre. Every point of the bisectrix of the angle with vertex at P is equidistant from the sides PQ and PR. Also every point of the bisectrix of Q is equidistant from the sides QR and QP. Hence its intersection I is simultaneously equidistant from the three sides, that is, I is unique and it is the centre of the circle inscribed into the triangle. In order to calculate the equation of a bisectrix, we take the sum of both unitary vectors of adjacent sides:

The incentre I is the intersection of the bisectrix passing through P, whose directing vector is u, and that passing through Q, with directing vector v:

I = P + k u = Q + m v                   k, m real

Arranging terms we find PQ as a linear combination of u and v:

k u - m v = Q - P = PQ

The coefficient k is:

Since all outer products are equal because they are twice the triangle area, this expression is simplified:

The centre of the inscribed circle is:

By taking common denominator and simplifying, we arrive at:

For example, let us calculate the centre of the circle inscribed in the triangle with vertices:

P = ( 0, 0 )        Q = ( 0, 3 )              R = ( 4, 0 )

| PQ | = 3               | QR | = 5                      | RP | = 4