Geometry with Trigonometry
100 Cartesian coordinates; applications Ch. 6 respectively, and soW 3 has coordinates u 3 = x 3 (u 1 −x 2 )−x 1 (u 1 −x 3 ) x 3 ...
Sec. 6.7 Coordinate treatment of harmonic ranges 101 6.4 If the fixed triangle[Z 1 ,Z 2 ,Z 3 ]is isosceles, with|Z 1 ,Z 2 |=|Z 1 ...
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7 Circles; their basic properties Hitherto our sets have involved lines and half-planes, and specific subsets of these. Now we i ...
104 Circles; their basic properties Ch. 7 P=M O l P Q M O (i) Suppose that|O,M|=k,so thatMis a point of the circle. We write ...
Sec. 7.2 Properties of circles 105 Similar results hold whenX∈[M,Q, that is the points of[M,Q]{Q}are interior to the circle whil ...
106 Circles; their basic properties Ch. 7 P S Q O Figure 7.2. P S Q O M P S Q O U Proof. (i) By 5.2.2 |∠OSP|◦+|∠SPO|◦+ ...
Sec. 7.3 Formula for mid-line of an angle-support 107 Proof.Letlandmbe the perpendicular bisectors of[B,C]and[C,A], respectively ...
108 Circles; their basic properties Ch. 7 With the notation of the last result, let Q≡F( 1 , 0 )and sl(Q)=P 3 where P 3 ≡F (a 3 ...
Sec. 7.4 Polar properties of a circle 109 By Pythagoras’ theorem, |O,T 1 |^2 =|O,U|^2 +|U,T 1 |^2 =x^2 +y^2 = a^4 b^2 +a^2 − a^4 ...
110 Circles; their basic properties Ch. 7 Thus as we we may replaceaandbbya/ √ a^2 +b^2 andb/ √ a^2 +b^2 , without loss of gener ...
Sec. 7.4 Polar properties of a circle 111 and this is fixed. We continue with the case wherePis exterior to the circle, and may ...
112 Circles; their basic properties Ch. 7 T 2 ∈[T 1 ,U. Similarly every point ofCis also in the closed half-plane with edge PT 2 ...
Sec. 7.5 Angles standing on arcs of circles 113 and of course (c) is ruled out by assumption. S R P O Q U P Q ...
114 Circles; their basic properties Ch. 7 of the reflex- angle with support|∠POQ. By addition we then have that the degree- meas ...
Sec. 7.6 Sensed distances 115 account. It can have negative as well as positive and zero values and it is related to the distanc ...
116 Circles; their basic properties Ch. 7 From these we conclude that Z 1 Z (^2) ≤l=(s 2 −s 1 )|W 0 ,W 1 |. (7.6.4) In particula ...
Sec. 7.6 Sensed distances 117 7.6.2 Sensed products and a circle...................... The conclusion of 7.4.2 can be strengthen ...
118 Circles; their basic properties Ch. 7 7.6.3 Radicalaxisandcoaxalcircles .................... In 7.4.2 our proof showed that ...
Sec. 7.6 Sensed distances 119 On subtracting the second of these from the first, and simplifying, we find that their radical axi ...
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