Let O and P be the circumcentre and orthocentre respectively. Draw OD and

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 circumcentre 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 circumcentre and the orthocentre.

L. T.

16

If OF be perpendicular to AB, we have

Also

<OAF 90° - ZAOF=90° - C.
=

< PAL=90° - C.

.. ZOAPA-ZOAF-PAL

=A-2 (90° C)=A+2C-180°

=A+2C (A+B+C)=C-B.

Also OAR, and, by Art. 209,

2

PA = 2R cos A.

.. OP2=OA2 + PA2 — 20A. PA cos OAP

= R2 + 4R2 cos2 A - 4R2 cos A cos (C – B)

=

cos (C – B)]

· R2 - 4R2 cos A [cos (B + C) + cos (C – B)]

= R2 - 8R2 cos A cos B cos C.

.. OP=R√1 – 8 cos A cos B cos C.

(Art. 72),

*217. To find the distance between the circumcentre

2

.. OI2= OA2 + AI2 – 20A. AI cos OAI

1

In a similar manner it may be shewn that, if I be the centre of the escribed circle opposite the angle A, we shall

Also, if & be the length of AD and 0 the angle it makes with BC, we have

We thus have the length of the bisector and its

XXXVII.

If I, 11, 12, and I be respectively the centres of the incircle and the three escribed circles of a triangle ABC, prove that

If I, O, and P be respectively the incentre, circumcentre, and orthocentre, and G the centroid of the triangle ABC, prove that

12. 102= R2 (3 – 2 cos A - 2 cos B – 2 cos C).

17. Prove that the distance of the centre of the nine-point circle from

18. DEF is the pedal triangle of ABC; prove that

(1) its area is 2S cos A cos B cos C,

and (3) the radius of its incircle is 2R cos A cos B cos C.

19. 010203 is the triangle formed by the centres of the escribed circles of the triangle ABC; prove that