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103. Through A, B, C the lines A, B1, B,C1, C24, are drawn perpendicular respectively to the sides AB, BC, CA of ABC: show that the circumradius of A,B,C, is to that of ABC as

sin? A + sin? B + sino C is to 2 sin A sin B sin C.

104. If the A-escribed circle is equal to the circumcircle, cos A = cos B+ cos C.

105. As any point P moves on the circumcircle, the quantity AP2. ABSC + BP. ACSA + CP2. AASB remains constant.

+

106. If from the primitive triangle a tangent to the incircle parallel to BC cut off a new triangle, and from the triangle so formed another triangle is similarly cut off, show that the aggre

Δs2 gate areas of all the triangles is

a (b+c)
107. If IS = 10, one of the angles of the triangle is 60°.

108. The distance between two points is a, their distances from a given straight line are b and c. Of all the triangles that can be described having the same base a and vertex lying in the given straight line, the area of that which has the greatest vertical angle is a J(bc).

109. If S, be taken on AS, so that SS, is cut by BC in the same ratio as it is cut at A, then

(1) AS, = R(say) = R tan B tan C.
(2) SS, = R cos A sec B sec C.
(3) S,S3, SzSı, S,S, pass through A, B, C respectively.

(4) The perpendiculars from S, upon the sides are as those from S.

(5) The intercepts on the sides by the circle with centre $, and radius R, are to one another as the sides.

(6) The tangents to this circle where it cuts the sides are antiparallels to the sides.

1 1 1 1
(7)
RR,

R, R2 R

=

+

+

ах

2

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110. If x, y, z are the perpendiculars from any point upon the sides of ABC, then (1) If the point is within the triangle,

ax + by + cz = 24. (2) If the point is within the space formed by producing AB and AC,

by + CZ :24. (3) If the point is within the space formed by producing BA and CA,

ax by - cz = 2A. 111. Show that four points may be found so that » : y : 2 is in any given ratio: and that the line joining any two out of the four points passes through a vertex of the triangle and is cut at the vertex in the same ratio as by the opposite side.

112. If J, J1, J, J, be the four points in the last example, viz. such that AJJ, J,AJ3, &c., are straight lines, and if through J lines B'C", C'A", A'B" be drawn parallel to BC, CA, AB respectively and cutting the sides in the hexagon A'B'C'A"B"C", then A'A", B'B', "C" will be parallel to JJ3, JJ, JJ, respectively.

113. In the last example, show that four circles with centres J, J1, J2, J, and centres of similitude A, B, C

may

be described all of which will cut the sides at the same angles; and that their radii Pe Pi, P2, P3 will be such that

1 1 1 1

+ +

р Pi P2 P3 [Particular cases of propositions 111, 112, 113 are given by 11,1,13, KK,K,K3, and SS,S,S3. (Ex. 109.)]

114. If J be any point, ABC any triangle, and if a . AJ= X, b. BJ= Y, C.CJ= Z, and abc = T, then

c
X? + Y? + Zo + T2

= 2 cos A (YZ + TX) + 2 cos B (ZX + TY) + 2 cos C (XY + TZ).
Hence find the radius of the circle circumscribing ABC.
J. T.

12

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a

.

=

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115. The distance from A of the point at which the sides of A BC subtend angles a, ß, y is

2A sin a + } (a62 c*) cos a {{sin a sin ß sin y (4A - a-cot a 62 cot ß-ca cot y)}* 116. If x, y, z be perpendiculars from any point upon the sides of a triangle, (x2 + y2 + z) (a+ b3 + c^) is never less than 4A?, and has its minimum value at K.

117. If through any point P within a triangle AP, BP, CP cut the sides in A', B', C', then

BC.CA'. A B'=CB'. BA'. AC'.
Also if BA' : CA' = n : m and CB' : AB' =l: n, then
AA'B'C' : AABC = 21mn : (m + n) (n + 1) (2+ m).

2Amn
And AP

where a, b, y are the (1 + m + n) l sin ß sin y' angles subtended by the sides at P.

118. If the perimeter of the triangle formed by the feet of the perpendiculars from any point P on the sides is p, then the least value of PA? + PBP + PC? is

pa

sino A + sin? B + sinC' and for this least value PA : PB : PC b

119. The area of the triangle formed by the feet of the perpendiculars from any point P within A BC is less than } the area of ABC by

}(AP?. sin 2A + BP?. sin 2B + CP2. sin 2C). 120. Lines B'C', C'A', A'B' are drawn parallel to the sides BC, CA, AB at distances x, y, z respectively; find the area of A'B'C".

If eight triangles be so formed the mean of their perimeters is equal to the perimeter of ABC, but the mean of their areas exceeds its area A by

aʻx2 + bạy2 + cz2

+

+

=a:

: C.

+

[In the following examples (121-124), ABCD is a quadrilateral inscribable in a circle.]

121. If BA, CD be produced to meet in 0, and if AO = x, DO = y, show that w and y are given by the equations

y d

d 7

(c+y); y = ž (a + 2).

X =

Hence find the area of ABCD from the formulæ giving the areas of OAD and OBC.

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123. If A, B, C, D be the areas of the triangles whose bases are a, b, c, d and vertex the intersection of the diagonals, then A B С D

Q
a. bd
ca. db

(bc + ad) (ab + cd)

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d, ac

=

124. If BA, CD intersect in E; BC, AD in F; and AC, BD in G, then

area EFG : area ABCD=2abcd, : (do ~ 5) (~ a). 125. Find the ratio of the perimeter of a regular polygon to the diameter of its circumcircle when the polygon has 6, 8, 10, 12, 20 sides. Evaluate the surd expressions in each case to four decimal places.

126. Show that the square described about a circle is of the dodecagon inscribed in the circle.

127. If R and r be the radii of the inscribed and circumscribed circles of a polygon of n sides, each = a,

90° = cot

2

a

R + r =

n

128. The area of an irregular polygon of an even number of sides described about a circle is equal to the radius x the sum of every alternate side.

129. If a, b, c, d be the sides of a quadrilateral circumscribing a circle, and if B, y be the angles contained by a, b and by c, d respectively, d

ab sin? 1 B = cd sin }y. Hence show that the area of the quadrilateral is

(abcd). sin } (B + y).

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