Advanced High-School Mathematics

SECTION 1.3 Circle Geometry 31 Let 4 ABChave circumradiusR. Show that Area 4 ABC= R(acosA+bcosB+ccosB) 2 , wherea=BC, b=AC, an ...

32 CHAPTER 1 Advanced Euclidean Geometry 1.3.2 Steiner’s theorem and the power of a point Secant-Tangent Theorem. We are given t ...

SECTION 1.3 Circle Geometry 33 Corollary. (Steiner’s Theo- rem)We are given the a circle, and secant lines(PA)and(PC), where (PA ...

34 CHAPTER 1 Advanced Euclidean Geometry Prove the “Explicit Law of Sines,” namely that if we are given the tri- angle 4 ABCwit ...

SECTION 1.3 Circle Geometry 35 Prove Van Schooten’s theorem. Namely, let 4 ABC be an equilat- eral triangle, and letC be the ci ...

36 CHAPTER 1 Advanced Euclidean Geometry Theorem.The quadrilateralABCDis cyclic if and only if ABĈ +CDÂ = CAB̂ +BCD̂ = 180◦.(1 ...

SECTION 1.3 Circle Geometry 37 Theorem. The points X, Y, and Z, constructed as above are colin- ear. The resulting line is calle ...

38 CHAPTER 1 Advanced Euclidean Geometry Proof. Whether or not the quadrilateral is cyclic, we can con- struct the point E so th ...

SECTION 1.3 Circle Geometry 39 proving the first part of Ptolemy’s theorem. Assume, conversely, thatABCDis not cyclic, in which ...

40 CHAPTER 1 Advanced Euclidean Geometry Exercises [AB] and [AC] are chords of a circle with centerO. X andY are the midpoints ...

SECTION 1.4 Harmonic Ratio 41 set A;X;B = AX XB (signed magnitudes); if A;X;B > 0 we call this quantity the internal division ...

42 CHAPTER 1 Advanced Euclidean Geometry LetA, B, X, andY be colinear points. Define thecross ratioby setting [A,B;X,Y] = AX A ...

SECTION 1.5 Nine-Point Circle 43 ratio. Show thatAB is the harmonic mean ofAX andAY.^9 The figure to the right depicts two circ ...

44 CHAPTER 1 Advanced Euclidean Geometry we need only recall that not all quadrilaterals are cyclic. Yet, as we see, if the nine ...

SECTION 1.5 Nine-Point Circle 45 (XZ) and (AC) are parallel as are (PX) and (BY′). Therefore,∠PXZ is a right angle. By the theor ...

46 CHAPTER 1 Advanced Euclidean Geometry Given 4 ABC, let Obe its orthocenter. LetC be the nine-point circle of 4 ABC, and letC ...

SECTION 1.6 Mass Point Geometry 47 and soDF :FA= 3 : 14. Intuitively, what’s going on can be viewed in the following very tan- g ...

48 CHAPTER 1 Advanced Euclidean Geometry Applying the converse to Menelaus’ theorem to the triangle 4 PQS, we have, sinceT, W, a ...

SECTION 1.6 Mass Point Geometry 49 The pointF is located at the center of mass—in particular it is on the line segments [AD] and ...

50 CHAPTER 1 Advanced Euclidean Geometry Example 1. Show that the medians of 4 ABCare concurrent and the point of concurrency (t ...