Advanced High-School Mathematics
SECTION 1.2 Triangle Geometry 11 the triangle. When these three points are collinear, the line formed is called atransversal. Th ...
12 CHAPTER 1 Advanced Euclidean Geometry Case 1. We assume first that X, Y, andZare collinear and drop altitudesh 1 , h 2 ,andh ...
SECTION 1.2 Triangle Geometry 13 1.2.5 Consequences of the Ceva and Menelaus theorems As one typically learns in an elementary g ...
14 CHAPTER 1 Advanced Euclidean Geometry Orthocenter. In the trian- gle 4 ABC lines (AX),(BY),and (CZ) are drawn so that (AX) ⊥ ...
SECTION 1.2 Triangle Geometry 15 property of the incenter will be given in Exercise 12 on page 153.) However, we shall proceed b ...
16 CHAPTER 1 Advanced Euclidean Geometry (⇒). Here we’re given thatABBC =APPC.Let P′be the point determined by the angle bisecto ...
SECTION 1.2 Triangle Geometry 17 Given 4 ABC and assume thatX is on (BC),Y is on (AC) and Z is on (AB). Assume that the Cevians ...
18 CHAPTER 1 Advanced Euclidean Geometry of the triangle 4 ABC. (This is because the circumscribed circle containingA, B, andC w ...
SECTION 1.2 Triangle Geometry 19 In the figure to the right, three cir- cies of the same radius and centers X, Y andZare shown ...
20 CHAPTER 1 Advanced Euclidean Geometry Step 1. Locate the pointGon the lines (AE) and (FB); we shall analyze the triangle 4 GH ...
SECTION 1.2 Triangle Geometry 21 [DXB], so GX XI ID DH HB BG =−1. [AY F], so GA AI IY Y H HF FG =−1. [CZE] (etc.) [ABC] (etc.) [ ...
22 CHAPTER 1 Advanced Euclidean Geometry The diagram to the right shows three circles of different radii with centers A, B, and ...
SECTION 1.2 Triangle Geometry 23 1.2.6 Brief interlude: laws of sines and cosines In a right triangle 4 ABC, whereĈ is a right ...
24 CHAPTER 1 Advanced Euclidean Geometry Law of Sines. Given triangle 4 ABCand sidesa, b,andc, as in- dicated, we have sinA a = ...
SECTION 1.2 Triangle Geometry 25 Proof. Referring to the dia- gram to the right and using the Pythagorean Theorem, we infer quic ...
26 CHAPTER 1 Advanced Euclidean Geometry In the triangle to the right, show that c = √ 1 +i+ √ 1 −i √ 42 (where i^2 =−1) 1 1 c ...
SECTION 1.2 Triangle Geometry 27 cosθ = b^2 −s^2 −p^2 2 ps . Equating the two expressions and noting thata=r+seventually leads t ...
28 CHAPTER 1 Advanced Euclidean Geometry Let 4 ABC be given with sidesa= 11, b= 8,andc= 8. Assume thatDandEare on side [BC] suc ...
SECTION 1.3 Circle Geometry 29 Proof. We draw a diameter, as indicated; from the above lemma, we see that θ 1 +ψ = 90. This quic ...
30 CHAPTER 1 Advanced Euclidean Geometry Exercises In the diagram to the right, the arc AB ̆has a measure of 110◦ and the measu ...
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