By Shang-Ching Chou; Xiao-Shan Gao; Jingzhong Zhang

This quantity includes a suite of twenty written types of invited in addition to contributed papers offered on the convention held from 20-24 may perhaps 1996 in Beijing, China. It covers many components of good judgment and the principles of arithmetic, in addition to computing device technological know-how. additionally incorporated is a piece of writing by means of M. Yasugi at the Asian common sense convention which first seemed in jap, to supply a glimpse into the historical past and improvement of the sequence Pt. I. the speculation of computer facts. 1. Geometry Preliminaries. 2. the world approach. three. laptop facts in airplane Geometry. four. computing device evidence in reliable Geometry. five. Vectors and computer Proofs -- Pt. II. issues From Geometry: a set of four hundred automatically Proved Theorems. 6. subject matters From Geometry

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ABC V . andw. B c F Figure 1-11 Solution. IS 1. 14. ifu = v = w = -~ then we have ~:~~ = t. 2. 14, we have proved a stronger version of the Ceva theorem: lines AF, BD. and C E are concurrent if and only if S PQR = O. 16 4 Let L be the intersection of AB and CD, K the intersection of AD and BC, F the intersection of BD and KL. and G the intersection of AC and K L. Then LF KF LG = GK ' L F K G Figure 1-12 4This theorem is sometimes referred as the basic principle of the projective geometry. It can be used to define the hannonic sequence geometrically.

We will see later that this will lead to short proofs for many geometry theorems. 46 Chapter 1. 114 Let I, m, u, and v be Jour lines. Let K be a rotation such that K(l) We define L[u, v] + L[l, m] = L[u, K(m}] . II v. It is easy to check that the addition of full-angles is associative and commutative. The main properties of the full-angles are summarized as follows. Ql L[u, v] = L[O] if and only if U Q2 L[u, v] II v. lv. Q3 L[l] + L[l] = L[O]. + L[O] = L[u, v]. L[u, v] + L[l, m] = L[l, m] + L[u, v].

81 sin(300) + cos(o-? = 1. /3/2. 36 Chapter 1. Geometry Preliminaries Proof. 80, setting (t = f3 = 30°, we have sin(600) = 2 sin(300) cos(300). Since cos(300) = sin(90° - 30°) = sin(600), we have sin(300) = ~ . 1 So far, the angles are always positive. To represent the signed areas using trigonometric functions. we need to introduce the oriented angle. which is also denoted by L. 82 The oriented angle ~ABC is a real number such that (1) the absolute value onABC is the same as the ordinary angle LABC.