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1 亮立理路數利行路利不不理論連利數例理論來數不數例行利易連索連 ii

2 Constructing Dynamic Learning Environment Using Computer student : Liang-Wei Guo Advisor Jen-Chung Chuan Ming-Jang Chen Degree Program of E-learning College of Science National Chiao-Tung University Abstract Mathematic teachers often encounter the problem that lots of the content cannot be learned by students in an ocular way. Nowadays, computers and network can be used as the supplement to the traditional learning environment and solve the problem. The study will discuss the van Hiele five-phase learning model, the correlation between learning theory and CAL, and the examples of using computer software in mathematical learning. Teaching material for mathematics is developed from these theories. Then the feasibility and instances of mathematical instruction are discussed according to the software used. In chapter 3, we explore the problem of M-configuration in the triangle, Cabri Geometry and Cabri Java are adopted. Dynamic geometry can be easily presented on the website and guided learning can be correlated through appropriate hyperlinks. In chapter 4, we show how Euclidean Geometries and Non-Euclidean Geometries (The Upper Half-Plane Model and Spherical Geometry) are connected by the use of Cabri Geometry, Cabri Java and Cabri3D. Learners can enter the domain of Non-Euclidean Geometries from Euclidean Geometries through the presentation of related hyperlinks. All results of study are presented on iii

3 老老勵老識老識不度更老老老不吝識更老不吝論不論來龍勵祥崙更女勵論 iv

4 錄錄錄錄論理論數理論數索利理理不立例論 v

5 論參 vi

6 錄錄論數流索數索索索 vii

7 類利 viii

8 理理理 ix

9 x

10 論說 1.1 數不爛量練理識理行量力略念連利識來年力年數領力 1 力力論力力 2 3 數 4 數度力 92 弄論論弄索論念練索不不索更了力年數了利數來行數 NCTM 數 -- 不了數了力 NCTM 老了數念 NCTM,2000 NCTM 數連藍更年,, 20, 老, 力量 90 力裡裡 1

11 利利 Bishop(1989) 勵 Clements Battista(1992) 來 Hansen(1998) 21 行年力力索立不路數力行數 1.2 Cabri geometry Cabri3D 兩 van Hiele 理論理論行行索力參 van Hiele 理論 2 參路行利 Cabri Geometry Cabri3D 兩 2 Frontpage Cabri Java Cabri Geometry *.fig 若說說裡參數 Cabrijava.jar 錄 3 Frontpage 利參數 <embed src="*.cg3" width="600" 2

12 height="500" align="right"></embed> Cabri3D *.cg3 4 說 5 連 6 van Hiele 理論數理論 CAL M-configuration 利 Cabri3D 1-1 3

13 ( 83) 念領識理論領識念理論領識理論數論 2.1 理論 van Hiele van Hiele 理論 1957 年蘭數 Dina van Hiele-Geldof Pierre Marie van Hiele, (1) (2) (3) / (4) (5) 不連 0 4 來, : Burger & Shaughnessy, 1986 Crowley, 1987 Fuys, Geddes, & Tischler, 1988 Hoffer, 來, : Clements & Battista,1991 Mayberry, 1983; Usiskin, 1982;Wirszup,1976 van Hiele 說 1 1 (Visualization) 輪廓來類行不來異數 4

14 2 2 (Analysis) 識力念不來菱連菱行利 3 3 (Informal Deduction) 兩說行行論理不理 4 4 (Formal Deduction) 邏理立理理來理理理異異不 5 5 (Rigor) 不理立理立立不 van Hiele(1986) 說理論力了了 (Information) (Guided Orientation) 說 (Explication) 索 (Free Orientation) (Integration) (Information) 說料領易識參領例例 5

15 (Guided Orientation) 了更參論料來了念離說 (Explication) 念念理識來數索 (Free Orientation) 勵不念不更念了更不連利連來更 (Integration) 勵理識利數念來不不念來來識 6

16 2.1.2 數理論 CAL Jonassen(2000) 1 識 (learning from computer) CAI (1) 練練 (drill and practice) (2) (tutoring system) (3) (intelligent tutoring system) 2 識 (learning about computer) 3 識 (learning with computer) (Mindtools) (intellectual partners) 來列 (1) 識 (knowledge construction) (2) 索 (explorations) (3) (learning by doing) (4) (learning by conversing) (learning by reflecting) 來論行行論刺來行行識立理論行刺行沈 84 1 識 2 數 3 4 年度 5 7

17 料不 6 立勵更 7 練練理論理類路類識識識理識不易 Piaget 不歷歷 80 理歷裡類理索裡沈 84 1 識便度 2 量 (menu) (icon) 令 3 念不來 4 度不留 5 練利 6 連行理 (deep processing) 利留 (retention) 更 7 度 8 利 9 Paivio 理論索連立 10 (advance organizer) 識聯 8

18 理論力裡裡識老識老不行 2001 Black McClintock 1996 年 the Interpretation Construction Design Model (ICON model) 1 observations in authentic activities 離了來 Poth 更來利 2 interpretation construction 識不立念更老立力不類行來 3 識 contextualizing prior knowledge 立了歷識念立索識念 4 cognitive conflict 料念行念 5 cognitive apprenticeship 流立練老識識 6 collaboration 參老數識勵識說力老說念 Newton et al.,

19 7 multiple interpretation 不不理來不理論來不理論了理論力更更 8 multiple manifestations 理念念念理論來沈 84 1 勵識不識 2 行度利弄索識 Atkins Orientation activities 2 念念, 識 3 advice statements help facilities 略更率參理 4, 立見 5 compare and contrast observe and identify 6 來便索 7 來 symbolic representation of phenomena the real-life referent 8 例立 hypotheses explanations definitions 類 categories rules 10

20 WEB 理論力年來路識易路量料 WEB,,2003 Tyler (1949) model 來量來 Taba (1962) 更更 model 量行度 Web-based Web-based 1 了裡力來異利不不立 2 Web-based Relan Gillani Web-based 3 Web 不連路念連 Draves (2000) 連類 Critical, Important, Nice 行 Chou Lin (1998) 識更 4 hypertext 度索 Web 11

21 識不度不 next 念索念連量狀 5 Kearsley (2000) 力 6 量 Bugbee (1996) 來量 Harrison Bergen (2000) 列了讀了論論 7 行度量度 Web-based 量理來更立理行 2.2 數不易不量邏易數行理 van Hiele 若量行 Polya 略類 4 朗例行來利行來行行論例 12

22 流流例行更例類 2-2 數流流不易量易來說數類 1 2 數 3 不論 Dynamic Geometry System 念略不易度力利來利滑來不易量料 1 異了便理利利便利利利滑行來狀來行索 2 連狀連度量連來行 13

23 量數不利來更 3 不若狀不狀不不 4 錄錄歷錄來利省了更歷行歷利行 5 JAVA SCRIPT 路不更行不更 Geometer s Sketchpad Cabri Geometry cabri Cabri3D 來 3D 了 Cabri Geometry 行切不行 Cabri Geometry Cabri3D 兩來數 Computer Algebra System 數利數利數來 2D 3D 數數來數 14

24 數 Maple Mathematica 數數 1 利力率 π 若 l n l 2n n 2n 利 ( l ) 2 = 2 4 ( l ) 利 2 利數 2n n 2 π x 李 92 兩數 y = a y = log a x 數 GSP 數 0<a<0.066 兩數利 Maple 利數 Maple 利 a 利數數力 3 利來念來數 Maple 數易來 4 3D 立度數 Maple 立立立來利 VRML 路便 5 GIF Maple 數 frame GIF 便路利數立列了兩類數 Flash Powerpoint Excel 來數李利 Flashmx 15

25 GP2 利 Powerpoint Mathps 數見來行李 Mathematics Magazine proof without word Powerpoint 數 Excel 數數連數 16

26 M-configuration 來數例 Cabri Geometry Cabri Java 來 M-configuration 論利 van Hiele 立來行索 2004 年 The Intel International Science and Engineering Fair 兩 The M - Configuration of a Triangle 參 ABC M-configuration B-Ca-Aa-Ba-C C-Ab-Bb-Cb-A A-Bc-Cc-Ac-B 連 AAa BBb CCc X92 (triangle center) 切參 2003 年論 (A.Myakishev (2003), The M-Configuration of a Triangle, Forum Geometricorum, Volume 3, pp ) 利 Trilinear coordinates 行數度不錄數度行連識 3.1 利 3.2 行索度數行利行利不 Myakishev Kimberling 理 17

3.1 M-configuration ABC BC P AB Q AC R BQ= QP= PR= RC 狀 M B-Q-P-R-C M-configuration (A.Myakishev (2003), The M-Configuration of a Triangle, Forum Geometricorum, Volume 3, pp135 144.

27 3.1 M-configuration ABC BC P AB Q AC R BQ= QP= PR= RC 狀 M B-Q-P-R-C M-configuration (A.Myakishev (2003), The M-Configuration of a Triangle, Forum Geometricorum, Volume 3, pp ) 3-1 M-configuration (1)Trilinear coordinates of a point Define P P ABC D E F PD = kα PE = kβ PF = kγ P Trilinear coordinate P(α,β,γ) P=α:β:γ Theorm P=α:β:γ Trilinear 離 kα,kβ,kγ k = 2σ / (aα+bβ+cγ) σ ABC (Kimberling,1998) 3-2 Trilinear coordinates 18

3-1 incenter centroid α:β:γ 1:1:1 α:β:γ 1 1 1 : : a b c csca : cscb : cscc circumcenter α:β:γ cosa : cosb : cosc orthocenter α:β:γ seca : secb : excenter A-excenter 例 secc α:β:γ -1:1:1 3-3

28 3-1 incenter centroid α:β:γ 1:1:1 α:β:γ : : a b c csca : cscb : cscc circumcenter α:β:γ cosa : cosb : cosc orthocenter α:β:γ seca : secb : excenter A-excenter 例 secc α:β:γ -1:1:1 3-3 (2)Homogeneous Barycentric Coordinates Define P P ABC Homogeneous Barycentric Coordinates 數 (α,β,γ) α β γ PBC PCA PAB 3-4 Homogeneous Barycentric Coordinates Homogeneous Barycentric Coordinates 3-2 Homogeneous Barycentric Coordinates incenter centroid circumcenter a : b : c 1 : 1 : a ( b + c a ) b ( c + a b ) c ( a + b c ) 19

3.1.2 Conjugates (1) Isogonal conjugate Define ABC 不 P 連 AP BP CP P isogonal conjugate 3-5Isogonal conjugate 例 3-3 Isogonal conjugate circumcenter incenter centroid

29 3.1.2 Conjugates (1) Isogonal conjugate Define ABC 不 P 連 AP BP CP P isogonal conjugate 3-5Isogonal conjugate 例 3-3 Isogonal conjugate circumcenter incenter centroid Isogonal conjugate orthocenter incenter symmdian 3-6 Isogonal conjugate Locus of isogonal conjugate 1 Q ABC 2 P OQ 3 P ABC P P Q locus 3-7 Locus of isogonal conjugate 20

points Isotomic conjugate 3-9 Gergonne Nagel Isotomic Locus of isotomic conjugate 1 Q

30 (2)Isotomic conjugate Define ABC M1 M2 M3 不 P 連 AP BP CP Q1 Q2 Q3 Q1 Q2 Q3 M1 M2 M3 Q1 Q2 Q3 連 AQ1' BQ2' CQ3' P Isotomic conjugate 3-8 Isotomic conjugate 例 Gergonne Nagel points Isotomic conjugate 3-9 Gergonne Nagel Isotomic Locus of isotomic conjugate 1 Q ABC conjugate 2 P OQ 3 P ABC isotomic conjugate P P Q locus 3-10 Locus of isotomic conjugate 21

31 (3)Harmonic conjugates Define W X Y Z 若 WY izx = WZ iyx Y Z W X 兩 Harmonic conjugates W Y X Z Y Z W X 兩 Harmonic conjugates 1 不 X 連 X ' Y X ' W X ' X 3-11 Harmonic conjugates 2 X ' Y W 連 XW ' WW ' X ' W X ' X Y Z 3 連 Y' Z ' WX Z (4)Ceva conjugate Define Cevian triangle of a point 1 P 不 ABC 連 AP BP CP A B C 2 A B C ABC P Cevian triangle 3-12 Ceva conjugate 22

32 Define Anticevian triangle of a point 1 P 不 ABC 連 AP BP CP A B C 2 P A A 兩 Harmonic conjugates A 理 B C 3 A B C ABC P Anticevian triangle 3-13 Anticevian triangle of a point Define P1-ceva conjugate to P2 1 P1 P2 不 ABC P1 P2 2 A1B1C1 ABC P1 Cevian triangle A2B2C2 ABC P2 Anticevian triangle 3 連 A1A2 B1B 2 CC 1 2 P3 P1-ceva conjugate to P P1-ceva conjugate to P Perspective triangles Define 兩異 A1B1C1 A2B2C2 若 AA 1 2 B1B 2 CC 1 2 P A1B1C1 A2B2C2 Perspective triangles P center of perspective AB 1 1 A2B2 C3 A1C 1 A2C2 B3 B1C 1 B2C2 A3 perspectrix 3-15Perspective triangles 23

33 3.2 Construction M-configuration 索 Polya 略類 4 朗例行 ABC B BC BC P C P BC BQ= QP= PR= RC BQ= QP= PR= RC P BP: PC = cos B: cosc 3-16 索 1 利 cabri java 3-17 索 2 24

34 更 3-18 索 M-configuration 索 M-configuration (1) 1 A BC H 2 BC A 3 連 AH' ' BC P 4 BP CP AB AC Q R 5 P Q R 兩 3-19M-configuration 1 ABC O H AE BAD CAH EAD EAH CLEAR 3-20 Isogonal conjugate1 25

35 ABC H H H H A B C CLEAR 1 連 AO E 連 BE CE O 3-21H A B C BE: EC = cos C: cos B 2 BAH CAE BH ' = CE BC// H' E 3 BC// H' E BE: EC = cos C: cos B= CH': BH' 4 A BAC BH A CH A 3-22 M-configuration 1 BH A CH A BE: EC = cos C: cos B= CH': BH' BP: PC = cos B: cosc 26

36 (2) H 行 BAC BC P BP CP AB AC Q R B-Q-P-R-C ABC M-configuration 3-23 M-configuration 2 1 O BC 行 L 2 BAC AA" L A' H ' H H BC 3 A' H ' BC P 連 PH 4 A AH = AH A PH A= A HA 3-24 M-configuration 2 A AH = PH A PH // AA " 27

37 (3) BAC AD O D 連 BD CD OQ // BD OR //CD AB AC Q R R BC L C L P B-Q-P-R-C ABC M-configuration 3-25 M-configuration 3 (Alexei,2003) 1 M BC Y Z Ca Ba BC 1 2 OM = a cot A = la (cos B + cos C )cot A 2 CaY = lasin B 1 MY = a lacos B= lacosc 2 3 CaOX CaY OM sin B (cos B + cos C)cot A = MY cosc sin Bsin A (cos B+ cos C) cos A = coscsin A cos( A+ B) cosccos A = coscsin A cos C(1 cos A) 1 cos A A = = = tan coscsin A sin A M-configuration 3 CaOX MBA CaO // BA' 理 BaO // CA ' 28

38 (4) 類 M-configuration ABC B C M-configuration 1 BC A 2 連 A AH' ' BC Aa1 M-configuration B-Ca1-Aa1-Ba1-C 3 連 A" H' BC Aa2 4 Aa2C Aa2B AB AC Ca2 Ba2 5 B-Ca2-Aa2-Ba2-C M-configuration Remark 3-27 類 M-configuration 2 2 2: 1 = (cos cos ):1 BCa BCa B C clear L L B C P 兩 BQP PRC BQ= QP= PR= RC

39 1 BC L1 L1 O 2 O BO O L1 A 3 連 AP ' O H 4 H L2 BC O A 5 連 AB AC BP CP AB AC Q R BQP PRC 2 不 B C Q 4BQ > BC 兩 BQP PRC P BC BQ= QP= PR= RC Q BQ BC P 2 C BC CP R 3 BQ= QP= PR= RC

40 3 不 B Q R M-configuration B-Q-P-R-C Q BQ QR P 2 連 BP R BP L 3 P L C 4 B-Q-P-R-C M-configuration

41 3.3 M-configuration 了 Cabri B-Ca-Aa-Ba-C ABC M-configuration ACaBa a ABC O b AaBaCa Ha a b 令 Ha AaBaCa. BaHaCa = π BaAaCa = π BAC = π CaABa. Ha ABaCa AaBaCa, BaHa REMARK a = CaHa Ha A ACaBa BCbAb CBcAc ABC O 3-34 M-configuration proposition M-configuration proposition2 REMARK b AaBaCa AbBbCb AcBcCc Ha Hb Hc ABC I 3-36 M-configuration proposition3 32

42 c AaBaCa Oa O H 離 3-37 M-configuration proposition4 REMARK c AaBaCa AbBbCb AcBcCc Oa Ob Oc OH CLEAR 3-38 M-configuration proposition5 若 BCaAa Pa,b CBaAa Pa,c BPa, b CPa, c 兩 Pa 理 APb, a CPb, c 兩 Pb APc, a BPc, b 兩 Pc 3-39 M-configuration proposition6 33

43 d PaPbPc ABC P 3-40 M-configuration proposition7 3.4 M-configuration 參 Kimberling(1998 X(92) 了 M-configuration 不 X(92) M-configuration B-Ca-Aa-Ba-C C-Ab-Bb-Cb-A A-Bc-Cc-Ac-B 連 AAa BBb CCc X(92) BCa = CaAa = AaBa = BaC BAa : AaC = cos B : cosc 理 CBb : BbA = cos C : cos A 3-41X(92) Kimberling,1998 ACc : CcB = cos A: cos B Ceva s Thm AAa BBb CCc X(92) CEVAPOINT OF INCENTER AND CLAWSON POINT X(92), Trilinear csc 2A : csc 2B : csc 2C X(92), Barycentric sec A : sec B : sec C X(92) = isogonal conjugate of X(48) X(92) = isotomic conjugate of X(63) 參 ( 34

44 Clawson point X(19): 1 Homotheic center of the orhtic triangle and extangents triangle( 切 ) 2 ABC 切 D E F G H I 六連 DE FG HI 兩兩 A' B' C' 連 AA' BB' CC' clawson point (Yiu, 2000) 3-42 Clawson point X(19) X(63) Isogonal conjugate of X(19) 3-43 X(63) Isogonal conjugate of X(19) X(92) Isotomic conjugate of X(63) 3-44 X(92) Isotomic conjugate of X(63) 35

45 利 Cabri3D van Hiele (Rigor) 不理立理不 van Hiele 理論兩 van Hiele 理論理理度來念若利更不易理若行更來更易不 91 Cabri uhpm.men Cabri3D 來兩理來 4.1 兩 4.2 兩切理理 Fermat 理立 line 兩 line x 4-1 line 1 PP 1 2 x c 4-2 line 1 兩 P1 P2 36

46 2 c cp1 x 2 Q1 Q2 QP 1 2Q2 2 segment 兩 line 兩 4-3 u-segment 3 triangle <180 度 >180 度 4-4 u-triangle 4-5 triangle 37

47 4 circle 兩 P1 P2 P1 hyperbolic center P2 4-6 u-circle 4-7 circle 1 P1 P2 x D 連 PD 2 1 兩 P1 P2 O 連 OP1 2 P1 x P2 PD 2 2 P2 OP1 circle 3 兩 C C CP 2 4 C P1 u-center P2 u-circle 38

48 5 u-center u-circle u-center 4-8 u-center 1 O MO x 2 PO MO P 3 P x Q Q PQ MO C C hyperbolic center (u-center) 39

x A B 3 APB 2 perpendicular from point 4-11 perpendicular

49 4.2 1 perpendicular at point 4-9 perpendicular at point 1 r1 m PC perpendicular at point r1 切 x C2 2 C2 PC2 r2 x A B 3 APB 2 perpendicular from point 4-11 perpendicular from point 1 令 P' P u- line m 4-12 perpendicular from point 40

50 2 P P' u- line n 3 perpendicular to two lines 兩 4-13 perpendicular to two lines 1. 令 C1 C2 u- lines m n B1C 1 B2C2 x 2 M B1 B2 B1N C1M B2N C2M 兩 N 4-14perpendicular to two lines 1 兩 line 兩 L 2 L 3 PN x P PC1 m Q 4 P PQ 41

51 4 two limiting parallel u-lines 行 4-15 two limiting parallel u-lines 1 A B m PA x C 2 令 A' A C APA' P m 行 u-line 3 B 行 5 two tangents to u-circle from point 切 4-16 two tangents to u-circle from point 1 O c Euclidean center A' A x 2 O A A' c S S' 4-17 two tangents to u-circle from point 1 A polar 2 L polar P Pt 42

52 3 AA' SS ' P OP c T T' 3 Pt L circle 兩 T1 T2 line PT1 PT2 切 T1 T2 切 4 T T' 切 u-lines AT AT' 切 6 Angle bisecter 4-18 Angle bisecter 1 B' AC' r 2 A r x R 3 A R u-line 4-19 Angle bisecter 1 A B C O O OA AC AB E F 2 A EF BAC 43

53 4.3 理 1 理 AB AB P 兩 CD EF 連 CF DE AB G H P GH 4-20 理 4-21 理 4-22 理 1 GI CD GJ EF HK EF HL CD 令 PA = PB = x 2 CGI~ EKH CG EH GI HK FGJ~ DHL GF HD GJ HL PGI~ PLH GI HL PG PH PGJ~ PHK GJ HK PG PH 4-23 理 3 PG 2 PH 2 GI HL GJ HK GI HK GJ HL CG EH GF HD ( AG * GB) ( BH * HA) ( x PG)( x + PG) ( x PH )( x + PH ) ( x PG ) ( x PH ) PG = GH Reference: L. Bankoff(1987), The Metamorphosis of the Butterfly Problem, Mathematics Magazine, v 60, issue 4,

2 Pappus Theorem L1 異 A1 A2 A3 L2 異 B1 B2 B3 A1B 2 A2B1 P1 AB 1 3 AB P2 3 1 A2B3 AB 3 2 P3 P1 P2 P3 4-24Pappus Theorem 4-25 Pappus Theorem 4-26 Pappus Theorem PQR B1P12A2 A1P13B3 A3P23B2 A1A2A3

54 2 Pappus Theorem L1 異 A1 A2 A3 L2 異 B1 B2 B3 A1B 2 A2B1 P1 AB 1 3 AB P2 3 1 A2B3 AB 3 2 P3 P1 P2 P3 4-24Pappus Theorem 4-25 Pappus Theorem 4-26 Pappus Theorem PQR B1P12A2 A1P13B3 A3P23B2 A1A2A3 B1B2B3 Menelaus 理 P12 P13 P23 Reference: Pascal Theorem 若 A1,A2,A3,B1,B2,B3 AB 1 2 A2B1 P1 AB 1 3 AB P2 3 1 A2B3 A3B2 P3 P1 P2 P Pascal Theorem 4-29 Pascal Theorem 4-30 Pascal Theorem Reference: 45

4 Desargues' Theorem O 兩 A B C ABC 若若 AB' ' AB C A' C ' AC B B ' C ' BC A A B C 4-31 Desargues' Theorem 4-32 Desargues' Theorem 4-33 Desargues' Theorem Reference: http://mathworld.wolfram.

55 4 Desargues' Theorem O 兩 A B C ABC 若若 AB' ' AB C A' C ' AC B B ' C ' BC A A B C 4-31 Desargues' Theorem 4-32 Desargues' Theorem 4-33 Desargues' Theorem Reference: 5 Radical center of the three circle 兩兩兩狀六 4-34 Radical center 4-35 Radical center 4-36 Radical center Reference: 6 Isogonal ABC 不 P 連 AP BP CP P Isogonal conjugate 46

4-37Isogonal conjugate 4-38Isogonal conjugate 4-39Isogonal conjugate 4-40 ex_isogonal

cut-the-knot.com/curriculum/geometry/conjugality.shtml 7 Isotomic http://mathworld.

html ABC M1 M2 M3 不 P 連 AP BP CP Q1 Q2 Q3 Q1 Q2 Q3 M1 M2 M3 Q1 Q2 Q3 連 AQ1' BQ2' CQ3' P

56 4-37Isogonal conjugate 4-38Isogonal conjugate 4-39Isogonal conjugate 4-40 ex_isogonal conjugate 4-41 ex_isogonal conjugate 4-42 ex_isogonal conjugate Reference: 7 Isotomic ABC M1 M2 M3 不 P 連 AP BP CP Q1 Q2 Q3 Q1 Q2 Q3 M1 M2 M3 Q1 Q2 Q3 連 AQ1' BQ2' CQ3' P Isotomic conjugate 4-43Isotomic conjugate 4-44Isotomic conjugate 4-45Isotomic conjugate Reference: 47

57 4.4 理不立例 1 Miquel ABC,, ABC Miquel circles,, ). Miquel's theorem, Miquel circles Miquel point 4-46 Miquel 4-47 Miquel 4-48 Miquel Reference: 2 ABC P D E F D E F 立 Reference: 48

3 Symmetry midpoint Lines of the same color are symmetrical with respect to a midpoint. 立 4-52 Symmetry midpoint 4-53 Symmetry midpoint 4-54 Symmetry midpoint Reference: http://sylvester.math.nthu.

58 3 Symmetry midpoint Lines of the same color are symmetrical with respect to a midpoint. 立 4-52 Symmetry midpoint 4-53 Symmetry midpoint 4-54 Symmetry midpoint Reference: 4 Fermat point ABC ACE BCD ABF 連 AD BE CF P Fermat point PA + PB + PC CF = BE = AD P 離 4-55 Fermat point 4-56 Fermat point 4-57 Fermat point Reference: 49

59 論不理論來行例 2003 Web-base 連不連利 Cabri Geometry Cabri Java 連行索 M-configuration 了裡更易行度利弄索識 Cabri Geometry Cabri Java Cabri3D 連行度念行索了參料連行了更來了不念立識行論 Cabri Geometry 利 Cabri Java Cabri Geometry 路行不論連念識立 Maple Mathemeical 來行不行行 Cabri3D Cabri Geometry 類立立行不行 50

60 立不不行利 Cabri3D 利利連兩不不利行更論 Cabri3D XML 行 1 數不 CPU 2 路不 Cabri Geometry 來便利類 Cabri Java Cabri3D 讀 3 Cabri Geometry 良 Cabri3D 錄流不利錄利 4 *.cg3 Internet explorer 易讀狀 Cabri3D 行更 2 3 路更易來數更易立數數易落路來行更量力了不力不易力浪若更數數更更 51

61 量行落 52

62 參 A. Myakishev (2003), The M-Configuration of a Triangle, Forum Geometricorum, Volume Botana F., & Valcarce J. L. (2002). A dynamic symbolic interface for geometric theorem discovery. Computers & Education 38 (2002) Chou, C., & Tsai, C. C. (2003). Developing Web-based curriculum: Issues and challenges. Journal of Curriculum Studies, 34(6), C. Kimberling (1998), Triangle centers and central triangles, Congressus Numerantium, C. Kimberling (1994), Central points and central lines in the plane of a triangle, Math. Magazine, C. Kimberling, Encyclopedia of Triangle Centers, May 23, 2003 edition available at Colette, L.(2000), Dynamic Geometry Environments as a Source of Rich Learning Contexts For The Complex Activity of Proving. Educational Studies in Mathematics, Kluwer Academic Publishers. Netherlands, 44: pp E. Roanes -Lozano & E. Roanes Macias.(2003). A bridge between dynamic geometry and computer algebra. Mathematical and Computer Modelling, George Polya 1953/1992 How to solve it 北 George Polya 1957/1993 Mathematics and Plausible Reasoning 李李理祿數北 G. L. EBERT(2001), Constructions in Finite Geometry Using Computer Algebra Systems, J. Symbolic Computation, Academic Press 31, pp55-70 Jean M. Laborde & Frank Bellemain(1998), Cabri Geometry II,Texas instruments, Software. Jean M. Laborde & Eric Bainville(2004),Cabri 3D, Cabrilog team, Software. P.Yiu (2000), The Clawson Point and Excircles, Unpublished. P.Yiu, Introduction to the Geometry of the Triangle, Summer 2001, version , Unpublished. Tsai, C.-C. (2001) The interpretation construction design model for teaching science and its applications to internet-based instruction in Taiwan. International Journal of Educational Development, 21 (5), 梁 90 力 van Hiele 六類立 87 pp 北 85 Cabri-geometre 數 ( ) 53

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