44. Let 12 be the Brocard Point at which AC subtends the supplement of A; and let a, B, y be such points taken in BC, CA, A B that the circumcircles of Asy, Bya, Caß cut in 2*. Then (1) aßy is similar to BCA. (2) 2 is one of the Brocard Points of aßy. (3) The Brocard angle of aßy is equal to that of BCA. (4) The angles Bla, CAB, ANy are equal (being the angles by which the figure BCA would have to rotate round 1 in order to take up the position of the similar figure aßy).

45. The cosine-centre is a Brocard Point of each of the triangles, similar to the primitive triangle, in which the Lemoine Circle cuts the sides :—the other Brocard Points of these triangles being the Brocard Points of the primitive triangle.

46. If X, Y are the feet of the perpendiculars from 0 on AB, AC; then XY is perpendicular to the antiparallel of 10

AO with respect to BAC.

47. The centre of the circle which passes through the feet of the perpendiculars from the Brocard Points on the sides is the point of intersection of the line joining the Brocard Points with the line joining the circumcentre and cosine-centre.

48. The Brocard Circle passes through the middle points of the three chords of the circumcircle which join the cosine-centre to the vertices.

* Thus: take a anywhere in BC, and let the circles NBa, 2Ca cut AB, AC respectively in y, B.




a =

C :

§ 1. CIRCLES AND TRIANGLES. 207. Expressions for the area (S) of a triangle.

By Art. 117 the area of a triangle =half the product of two sides and the sine of the non-obtuse angle included by them.

Now since the sine of an angle is equal to the sine of its supplement (Art. 137), the sine of the internal angle may be always used in this expression. Thus: given two sides (a, b) and the included angle (C),

Se area of triangle = }ab sin C...............(1). csin A

csin B Now

and b =

sin C .. substituting in (1),

c- sin A sin B S = area of triangle =


2 sin C This form gives the area in terms of the angles and one side. Again, by Arts. 150, 152, (1) becomes

S = area of triangle = /{8 (8 - a) (8-6)(8 - c)}......(3).
This form gives the area in terms of three sides.
208. Expressions for the radius (r) of the inscribed circle.
Bisect the angles at B and C by BI and CI.
Draw perpendiculars IX, IY, IZ on the sides.
Then IX=IY = IZ=r. Now

area of A BIC= 1 BC.IX = a.r,
area of a CIA = CA. IY = 16.r,
area of A AIB = ! AB. IZ= c.r,

:. (adding) area of A ABC = } (a + b + c) r = 8.7,

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Now AY = AZ; BZ = BX; CX=CY.

:: AY+BX+CX = } (a + b + c) = 8. .. AY or AZ=8-a; BZ or BX=8-b; CX or CY=8 - c.

:: p=IY= AY tan IA Y = (8 - a) tan : A.........(2). Or again :

BC sin IBC p= IX = IC sin ICX =

sin ICX

sin BIC
a sin 1 B sin 1C a sin B sin C
sin } (B+C)


cos A 209. Expressions for the radius () of the escribed circle. Bisect the exterior angles at B and C by BI, CI Draw the perpendiculars 11X1, 11Y, 1,2, on the sides. Then I,X = 1,Y,=1,2=r. Then

area of a BI C = BC.1,X = }a.ri,
area of A CIA = CA.I.Y. =,

area of A AIB=AB. I,2 =,
.. (subtracting BI C from CIA + AI,B),
area of - ABC = } (b+c-a) rı = (8 a) ,








.. Τι

8 - a



= 1

AY,= AZ; BZ, = BX,; CX=CY.
:. AZ = AB + BX, and AY,= AC + CX,.


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.. AY, or AZ= (a + b + c) = 8, and BX, or BZ, =s - c; CX, or CY, =s 6.

.. r = AY, tan I,AY, = 8 tan A.......

= BX, cot BI.X = (8 -c) cot B. Or again :

BC sin BCI rn = 1X, = BI, cos BI,X,

cos BI,X

sin BIC a cos i cos B a cos B cos C sin ? (B+C)


(3). 210. Expressions for the Radius (R) of the circumscribing circle.






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sin B

Through B draw the diameter BA' of the circumscribing circle. Join A'C.

Then the angle BCA' in a semicircle is a right-angle. And the angle BA'C, which is in the same or the opposite segment to BAC is either equal or supplementary to A.

.. sin BAC = sin A.

BC Now sin BAC =

.. sin A = BA''


6 .-. 2R=

(1). sin A

sin C .. d, which was used in Art. 145 for the value of each of these fractions, is the diameter of the circumscribing circle.


abc Now

sin A
.. R=

(2). The formulæ (1) may be also derived by using the figure of the circumscribing circle in the last chapter.

COR. The distances of S from the sides of ABC are R cos A, Rcos B, R cos C. If A is obtuse, cos A is negative, and the arithmetic value of the distance of S from BC is – R cos A, S being on the side of BC remote from A.

It will be found convenient to express distances in terms of R and ratios of the angles.

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