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IaqKT(b;4(t!GU3aKT(bY43
/->
Š ‹
I
V A G
=
c`Q; (5;4yŠJ‹X; (5;4yŠJ•‹8U3ŠJ‹_3nb
;;C!G;ŽYT/->
Nhận xét 1: Sử dụng bài toán 1 đã nêu với tư cách là tri thức phương pháp ta có
thể giải được các bài toán sau đây bằng phương pháp tương tự: Gắn điểm cần tìm
tập hợp điểm với một điểm nào đó đã biết tập hợp của nó qua một phép vị tự.
Nhận xét 2: Ở bài toán 1 hai điểm B, C cố định, điểm A di động. Bây giờ cho hai
điểm B, C di động, điểm A cố định ta có bài toán sau:
Bài toán 2: K(5;4yŠJ‹8(t!QT(b;4ŠJ‹aqK83(t!( 
;4(5;z#`!;RS;4;C!XU3;3!QaK
$588;;8/%A! ;^;Z@;RS;GU3
X'Y33(t!( aqK
2V:R†‡:;
+->
Š ‹
I
V I G
=
!8
I d
∈ ⇒
;RS(t!X];`!8(5;z•8U3(5
;z_3
+->
Š ‹

I
V I

c‰O;:;8+aqK( ;4(5;4yŠJ‹Y3C
aKA( 8%"(@q;3A8;Y3
Bài toán 3:K(t!QT(b;4(5;4y;C!ŠJ•ˆ‹qaqK83(t!( 
;4ŠJ•ˆ‹Y3aKA( 8%"(@#`!;RS;4;C!XU3;3!
QaK
2V:R†‡:;
$588;;8+8;>%AZ@];`!;RS;
GU3X7t;`!;RS(t!GYRP%:;
75%)(_3;4(t!U3Cq(b…#QXJsO;:;U3
8;qY43JGA( 8%"(@(t%:;R;RS(t!G
Bài toán 4:K(t!T(bo!8(5;4y;C!ŠJ•ˆ‹q;•%‘;:;:
Q8;;:aK;OŠJ‹#`!;RS;4;C!XU3;3!QaK%;
;:aK;3(@
2V:R†‡:;
XG8;4(t!U3aK#3A
+->
+
Š ‹
>
I
AI AG V I G
= ⇒ =
uur uuur
#3];`!;RS(t!Gq`G`(J%"(@O! ;A"G
; (5;4yŠJ•‹(5%)JX; (5;4yŠJ••‹8U3U3
ŠJ•‹_3nb;;C!Q;ŽYT+->
l ;%A%D!OYU38;8!88;;4O%"A8

OU3_•;)2:YY4R%";x8;;4O;`
%:;RU38;Y’Y337t:_•;)Y(t(:;4
(t!GU3aK†o!;4ŠJ‹G; Q•JQŠQ•83(t!;3
U3ŠJ‹8ŠJ•‹qY43X; QQ
.
8U3Q•JQ_3nb;;C!
Q;ŽYT+->
38;8^Y;p4o%";t!!A8
;3!8AY!;;:(TO!‚8;#•(Aqx!Y’
;p(SY(3^U38;q;p(S‘(‡;(t
x!;!;)A
Bài toán 5 K(5;4yŠJ‹qCQaT(b8l8! ;(t!( ;4
(5;4y8(t!Y3Qla8``8#`!;RS;4;C!XU3
;3!al%(t!l;3(@
2V:R†‡:;
QqaT(b
Š ‹ q Š ‹ Š “‹
AB
MP AB T M P M O P O
= ⇒ = ∈ ⇒ ∈
uuur
uuur uuur
8U3ŠJ‹_3n
;b;:;xx;Qa
l{;%
+->
+
Š ‹ Š ““‹
>
I

AG AP V P G G O
= ⇒ = ⇒ ∈
uuur uuur
8U3ŠJ•‹_3nb;;C!
Q;ŽYT+->
Bài toán 6:K(t!T(bo!8(5;4y;C!ŠJ•ˆ‹ql8! ;(t!
( ;4ŠJ‹68`:U3Jl
3‹ #`!;RS;4;C!XU3;3!lJ
‹ #`!;RS;4(t!GU36
‹ #`!;RS;4;C!X•U3;3!6J
2V:R†‡:;
3‹  J  T  (b  Y  43  ;4  (t!  ”  U3  J  T  (b   
/->
Š ‹
J
V M G
=
!8
Š ‹ Š “‹M O G O
∈ ⇒ ∈
8U3ŠJ‹_3
/->
J
V
‹6"`(JO! ;A"6; (5;4yŠJ
/
‹(5
%)Jc`6o!;4ŠJ‹O_•;)63%‘3;:;:
QqaOŠJ‹Y436; QJa
/-+

Š ‹
P
V H I
=
Y43;RS;4(t!GU368U3QJa_3nb;
;C!;ŽYT•
‹
/->
Š ‹ “
J
V H G
=
;RS;4;C!X•U3;3!6J8U3QJa
_3nb;;C!”;ŽYT/->
Nhận xét: Các bài toán trên đã có tỷ số k (Dễ thấy do tính chất của trọng tâm tam
giác hay trung điểm). Bây giờ ta nâng dần mức độ khó khăn bằng cách chưa cho
tỷ số k.
Bài toán 7:K(t!l( ;4(5;4yŠJ‹8! ;(t!GT(bo!
8ŠJ‹#`!_•;)3(t!U3GlO(5C;488
U3AGJl
2V:R†‡:;
7{;JG–•Jl–ˆ
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EI d EI d EI d d
IE IM
EM R EI EM d R EM R d d R
= ⇔ = ⇔ = ⇒ =
+ + + +
uur uuur
#;;3A

d
IF IM
d R
=
−
uur uuur
7{;
/ +
/ +
• Š ‹ • Š ‹
k k
I I
d d
k k V M E V M F
d R d R
= = ⇒ = =
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•Y43;RS(t!—q˜
#3A;tC]!( %AU38;o;D](t!;43
Kzq;`!;RS(t!l!8l8U3l
/
ql
/
8U3l
+

Bài toán 8:K3(5;4yŠJ‹8ŠJ•‹~;3;Q8aq;•! ;(t!l
( ;4ŠJ‹;3’(5;zlQqla~;ŠJ•‹;KqI#`!;RS;4;C!
6U3;3!QIK
2V:R†‡:;

7C8! ;8;%A( P%:;_3
75;z™$—q!T\m3t!;4;C!q;4;C!q;C!(5;4y
;:;3!lT;`!;RS(t!6;3;`!;RS;4;C!Xq!T
;`!;RS(t!X;3];`!;RS(t!GU3KI
#`!  ;R S(t!G#3 A 
·
·
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DAC AMD ADM
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+ +
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Š “q ‹C O R l R
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A
V I G
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X; (5;4yŠK
/

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O
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+
‹8
U3ŠK
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;4yŠK
+
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+
Š/ >‹ “ >Š/ ‹ . + “
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O J AJ AJ O J O
− + − = ⇔ − =
uuuur uuur r uuur uuuur ur
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Š#)+nb;
/
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k
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O
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/
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Š/ ‹ Š/ ‹k O O k O O
− = −
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‹
Nhận xét: Nói tóm lại việc xây dựng chuỗi bài toán nâng dần mức độ khó khăn
của nó trong quá trình dạy học không những tạo cho học sinh sự hứng thú trong
học tập mà còn giúp các em dễ dàng hơn trong việc tiếp thu tri thức mới.
QQJ%&'.€+{L)yJ
#4_;4`;q5]]Y;`!;yq
Y;PŠ;4(P%\A;t(S‹K3Y’^
YA;!P!‘q;q;Y;p(S;^;4
;3\;`!P5qY8(SUT%:;q8;
P!;8yY3!;x!#`!(SP58\;R
Y:;`Rp(P;xP%37P(A4p;@
)\;;4t;
#3wn;)Y3
Bài toán:K(5;4y;C!JqCQaT(bql8! ;(t!( ;4
ŠJ‹#`!;RS;4;C!6U3U3;3!lQa
2V:R†‡:;
E860;62k:R•6E•5ˆ:R6}•;
[n;;3!lQa;4;45SQ

#3  A 
·
·
.
/ /
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+ =
Š#       ;:‹  Y  43
·
·
.
/1.AHB AMB const
α
= − = =
qY436; 3A 
α
U3(
Qa
#;;45SAQ;s
E86Q;‰:R•6Š•5Y:654P:;
X  Q•  8  (t!  (T  w  O  Q  _3  J  %  (A
“ Š ‹
AB
MH A B v const T M H
= = = ⇒ =
uuur
uuuur uuuur r
q` 
Š ‹M o
∈ ⇒

6; ŠJ•‹8
U3ŠJ‹_3n;b;:;xx;Qa
E86U;‹†j:R5W:686r55W868ŒZ6Z4•6Š•p4ai:R5w?dN?q5•6Š•5Y:654P:J
Xl•8(t!(TwU3l_3Jq%(AQ6al•8``8Y43l•6
~;Qa;;4(t!GU3Qaq;3AJqGT(b
“ 
 
O
I
D
D
I O
M M H
D D M H
→ →
→
cR;RS68U3ŠJ‹_3;)3n(Tw;C!J8G
E86X;W868ŒZ6Z4•6Š•BY57D6E85w?;
XX8;4;C!U3;3!QlaY43Jq6qX;z88
>OH OG=
uuur uuur
q
;x  ;)  p;  ;4  ;C!  ;3    A
>/->
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/
 
>
O
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V
V
O I
IG IM M G H V V M H= ⇒ → → ⇒ →
uur uuur
cR;RS(t!8U3ŠJ‹_3;)3nb;A;4
E86\;‹†j:R•6Š•p4ai:R5oj8J
3J’ QaXl••8(t!(TwOl_3q;3A68(t!(Tw
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Y43;( (t!6
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J
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 ;Z;45@;"]wC(S\;T
8;_3(:;;€q;•(A;;^Yq;4;`;4
Y4o;8%\!;4•;Sw345;;:q8!!@
;RU3x!&3(Cq;"w(343! ;YT8;;;€(y}
;4;:;(t_:;
Bài toán:ŒY%3//;48n(Tw;4A)Y3K(5
;z83(t!Qqao!P! ;)3YO#`!;4! ;(t!lY3
;@QlšalA;4b}p;
4A4;
XQ•8(t!(TwOQ_3q%(AO!(t!l;4;3A
Qlšla–Q•lšla
cRQlšla}p;%Q•lšla}p;q!8Q•lšla}p;%8†
%Q•qlqa;z8%(Al83(t!U3Q•a8
Nhận xét: Việc giải bài toán này đơn giản, tuy nhiên sau khi giải xong bài này ta
có thể sử dụng bài toán này như là một tri thức phương pháp để giải các bài toán
liên quan đến thực tiễn sau đây:
Bài toán 1:KA3%8Zs)3(TO(543wxB3Š((543

88(5;z‹wC! ;83Zb;4)8(t;@(5(;•3
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2V:R†‡:;
šKt8;;;:8P8;;];;ŠK)88;Z
;4‹
šX8;;];;
š#458;;;€;•%:;_;(SZ;4
Bài toán 2:KA3%8Z3b;4)QqaZs)3(TO(543wxB3
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+.%!Zb;4)8(t;@(5(QKšKIšIa8}p;
2V:R†‡:
#`!t8;8P8;3(]on(Tw;4Y3
XQ•8(t!(TwU3Q_3qK8(t!p;%|;4qI8(t!; Y3
KI–+.
I``8KIa—%(AQKšKIšIa–Q•KšK—šKI}p;%
8†%Q•KšK—}p;q%(AK83(t!U3Q•—8Y43(t!I#•
(Aw(b(Sb;4)wC83
Bài toán 3:KA3;b;4pQqao!P3)3U3! ;Y"Š35wqt
;bo3(5;zYY‹6}wC]KI"AO3
5Y"Zb;4)8(t(5(;•Q(:a_3]8~p;
2V:R†‡:
KI%"(@QKšKIšIa}p;%8†%QKšIa}p;
I``8QKIQ•%(AQKšIa–Q•IšIa}p;%8†%
Q•qIqa;z8q;•(A;3A%:;_8;;;€
Bài toán 4:KA3;8TQqaqKZ>b;4);;8! ;;3!#4(5(
;•Q(:aA! ;%8q#4(5(;•Q(:KA! ;%8#`!;4
(5(;•a;OK! ;b;4)(t({;! ;;4;C!CT8Y3;@
(5(;•;4;C!;O3%88~p;
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2–l
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;4;C!CT8;
Bài toán 5:KA3;8TQqaqKZ>b;4);;8! ;;3!AAQ
#4(5(;•a(:KA! ;%8({;;6}wC;4;C!
CT8;l;4(5Qaq2;4(5QKY3;@)R
t}p;
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I
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lšl2š
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qlq2q
+
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Bài toán 6:KA3;8TQqaqKZ>b;4);;8! ;;3!#`!;4

(5(;•Q(:ab;4)wC;4;C!CT8;l;4(5;•Q
(:Kb;4)({;;4;C!CT828;4(5aKb;4)({;;4;C!
CT8Y3;@)Rt}p;
2V:R†‡:
#4aKp(t!p;%|q
/
(TwO_3Qaq
+
(Tw_3QK
9(Alš2š2}p;%8†%lq2; (5;z
/

+
c8lš2š2–
/

+
aC5;3;`!b;4)(t!;4aK(t
/

+
}p;
c`;3!Q
/

+
C;QA
·
·
/ +

+P AP BAC const
= =
Y43
/

+
}p;%
/ +
AP AP AP AP BC
= = ⇔ ⊥
$RR;;;3A
•AB BN AB CM
⊥ ⊥

cR(5p%^lq2q}p;%8†%lq2q8C(53
U3;3!QaK#•(AY43%:;RP8;;;:
QXJ+zLI;
/‹ K``8QaKIAQaT(bq(5nQKA( 8o!
%"(@9K;3(@q;`!_•;)(t!I
+‹ K(5;4y;C!J83(t!Qqal ;(t!l;3(@;4(5;4y
ŠJ‹#`!_•;)(t!l•Y3
MBMAMM =+“
>‹ K`;3QaKIA3(8Qa8KIK:;Q8aT(bq
QI–3qIK–Š3q8oYT‹#`!_•;)(t!I8K
41#4;3!QaKA3(†aqKT(byQ;4(5;4yŠJ•ˆ‹
T(b%"A(t!O(5aK#`!_•;);4;C!XU3;3!
QaK
\1K;3!QaKAaKT(bq(t!Q( ;4! ;(5;z
Y3aK%"~;(5;z#`!;RS
3‹ #4;C!XU3;3!QaK

‹ #4(t!GU3aK
_‹K(5;4yŠJ•ˆ‹(5%)QaT(bql28(5%)( K8
;4(t!U3%)JQ#`!;RS(t!83(t!U32K8a2
h1K(5;4yŠJ•ˆ‹8(t!T(bo!8(5;4yŠJ•ˆ‹l ;C
aK;3(@U3ŠJ‹A( 8%"(@o
>R
#`!_•;);4
;C!U3;3!aK
J/)yG).LJ
/0&3%;4t%3(P;8q]:;Y4p;;^O8;R8q
%:;_8x!(„:;R;:;(t;qx!AP;: q(3
YTYt8R;T;88;Rq;R!)m84p;;
7r;5qx!•;;`!;y43P8;(S
7{\;qx!(„:;_:;! ;\_8;;;€&3%
;B\!8(Tq;";(S%:;_Y3

7T
FV• 
^5SOn8>n 6<:Rp^5SOn8>n
 •  •
//Q+ ,< >> =/= // +1>
#B\!
FV• 
^5SOn8>n 6<:Rp^5SOn8>n
 •  •
//Q/ ,/ >0 0Eq/ + ,q0
JF)z;
2APU3n:`%"†A;"(„;4`
8;4(P;88q!8U3A8"s4 O#qO%"
%@U3(P;8•;);;€U3A;"†43! ;YT;4

#4mD!_3;"(„R;4(T;S
Y%}U3;45#6#789:;q;4(S;rƒY";
#28\;(3(z8rƒY}q;";p4oY
;:;;(TU( q(3YTYt8R;T;;4_;4`
8;RZ;4
 #4(C8! ;YTY8(Pwp;U3;"q!(AAs(r
\(t^(ƒY%3;;T;U3n:`;4
;4`;@;"8!YZ;3!3%|;Tp•
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Tân Phú, ngày 18 tháng 05 năm 2014

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