Plane trigonometry, by S.L. Loney.

CIRCUMCENTRE AND ORTHOCENTRE.

241

Let 0 and P be the circumcentre and orthocentre
respectively. Draw OD and
PK perpendicular to BC. A
Let AD and OP meet in G.
The triangles OGD and F
L
PGA are clearly equiangular.o. p
Also, by Art. 200,
B C
OD =RcosA B- K
and, by Art. 209,
AP= 2R cos A.
Hence, by Euc. vi. 4,
AG AP
GD OD '
The point G is therefore the centroid of the triangle.
Also, by the same proposition,
OG OD 1
GP AP 2'
The centroid therefore lies on the line joining the
circuncentre to the orthocentre and divides it in the ratio
1: 2.
It may be shewn by geometry that the centre of the
nine-point circle (which passes through the feet of the
perpendiculars, the middle points of the sides, and the
middle points of the lines joining the angular points
to the orthocentre) lies on OP and bisects it.
The circumcentre, the centroid, the centre of the
nine-point circle, and the orthocentre therefore all lie on a
straight line.
216. Distance between the circumnentre and the orthocentre.

L. T,

16