The notation adopted in this paper for the triangle ABC is:

G the centroid.

I the centre of the inscribed circle.

I1, I2, I3 the centres of the escribed circles within angles A, B, C.

O the centre of the circumscribed circle.

D, E, F; D1, E1, Fl; D2, E2, F2; D3, E3, F3 the points of contact of the inscribed and escribed circles with BC, CA, AB. The Ds lie all on BC, the Es on CA, and the Fs on AB.

H, K, L the mid points of BC, CA, AB.

X, Y, Z the feet of the perpendiculars from A, B, C, on BC, CA, AB.

A′, B′, C′ the vertices, opposite to A, B, C, of the triangle formed by drawing through A, B, C parallels to BC, CA, AB.