Plane Trigonometry by Sidney Luxton Loney

Author:Sidney Luxton Loney [Loney, Sidney Luxton]
Language: eng
Format: epub
Publisher: At the University press
Published: 1893-04-13T18:30:00+00:00

FIg.l. Fig. 2. Fig. 3.

The point 0 may either lie within the triangle as in Fig. I., or without it as in Fig, II., or upon one of the sides as in Fig. IIL

Taking the first figure, the two triangles BOD and COD are equal in all respects, so that ZBOD^ZCOD,

.'. z. BOD = J z BOG =z BAG (Euc. iii. 20),

BD^BOwiBOD.

Also

.'. -=^B^\iiA.

If A be obtuse, as in Fig. II., we have z BOD = i Z BOG = Z BLC = 180° - ^1 (Euc. iii. 22), so that, as before, sin BOD = sin il, and -B = tt—' —r •

2sinil

If -4. be a right angle, as in Fig. III., we have R^OA^OG^"^

a

2sinil

, since in this case sin A^l,

TRIGONOBiETRY.

The relation found above is therefore true for all triangles.

Hence, in all three cases, we have

a be

R =

SsinA asinB 28inC

(Art. 163).

201. In Art. 169 we have shewn that

where 8 is the area of the triangle.

Substituting this value of siuil in (1), we have

abc

R =

48

giving the radius of the circumcircle in terms of the sides.

202. To find ike value of r, the radius of the incirde of the triangle ABC,

Bisect the two angles B and C by the two lines BI and CI meeting in /.

By Euc. in. 4, / is the centre of the incircle. Join lA, and draw ID, IE and IF perpendicular to the three sides.

Then ID=IE:=^ IF^r.

We have

area of A IBC^\ID.BC^\r,a,

area of A ICA = \IE. CA =ir.6,

and areaof A J4B = 4/^^.-45 = Jr.c

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