The triangle \(ABC\) has vertices \(A(7, 7), B (0, 6)\) and \(C (6, 0)\). Write down the equation of the internal bisector of the angle \(BAC\).

Find, by an algebraic method,

  1. the equation of the circle through the points \(A, B\) and \(C\).
  2. the equation of the inscribed circle of the triangle \(ABC\), i.e. the circle which touches all three sides of this triangle and which has its centre within the triangle.

We will certainly need some clear and accurate diagrams with everything we know marked in.

Do we have any information as to where the circumcentre will lie? The incentre?

Can we find the coordinates of any key points? The lengths of any key sides?