1.Three circles are described each passing through the orthocentre of a triangle and two of its vertices ; show that the triangle formed by joining their centres is equal in all respects to the original triangle.

2.A triangle is inscribed in a circle, and any point P on the circumference is joined to the orthocentre of the triangle ; show that this joining line is bisected by the pedal of the point P.

3.I is the centre of a circle inscribed in a triangle, and I_1, I_2,I_3 the centres of the escribed circles ; show that II_1,II_2,II_3 are bisected by the circumference of the circle.