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YA;;T;!OC;)(Sq;•(A!O(t;
(S
+J€G#•);
Kn:`8U3A;4`GO//q
;;;€
J!L;
7P;8(Sq;B\!;4!%T//q;tqO//Q/q
//Q+;45#6#789:;
7T//Q+q;B\!//Q/
J$#•);
7t;\!()8\!U3(P;8;4_;4`;"(„
YBA!Y3
2;8\Y%3qYqY;3!%
I5q;43(@O(r\(tAP3
#43(@Ox!YP8;Pn:`q8
;;;:q;•(Apx!! ;O;T;
#\!8%t!;43
#4_;4`(P;8q;"(„;:8;\!O//Q/q//Q+
U3;45
J‚•.L;
0J$FHF)z;
9A(:3;4yqb;4)U3\8;R8wC\;T8;R8
Y!q8XJ$…Q(„Auc\;8! ; RU3
P_;4`qU3!;( ;A);4;45@;"u2~!
m!";q(A8ua:;;%"†8;R;";5!8m

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 ‹

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
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;b;:;xx;Qa
l{;%
+->
+
Š ‹ Š ““‹
>
I

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
—q˜8C(5C;488U3AlJG;3A
EI d EI d EI d d
IE IM
EM R EI EM d R EM R d d R
= ⇔ = ⇔ = ⇒ =
+ + + +
uur uuur
#;;3A

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 
·
·
·
DAC AMD ADM
= + =
oYT 
+DC l
⇒ =
Š9" (@‹
+ +
“O I l R
⇒ = −
%"(@G; (5;4y
+ +
/
Š “q ‹C O R l R
= −
#RS(t!X
+->
Š ‹
A
V I G
= ⇒
X; (5;4yŠK
/

Š ‹
O
V G H
=
#)U33nb;8! ;nb;
+
J
V
qO”(Sw(bY3
+
Š/ >‹ “ >Š/ ‹ . + “
>
O J AJ AJ O J O
− + − = ⇔ − =
uuuur uuur r uuur uuuur ur
Y43J•8;4(t!U3Q”
Š#)+nb;
/
/
k
O
V
8
+
+
k
O
V
8nb;
k

#3  A 
·
·
.
/ /
/1.B HA AMB
+ =
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
∈ ⇒

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
Olv_3QaY4368U3l_3;)3n(Tw;4Qa ;•(A
Y43;( (t!6
QUJ%&'.€+F#/)
J
#~;P;•;;€8(SwC(t;;€c`Rq%
 ;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

/
(TwOl_3aKq%(Alš
2–l
/
š2}p;%8†%l
/
qq2;z8q;•(AY43b;4)({;
;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;
2V:R†‡:
#@)Rt}p;%lš2š2}p;
I
/
(TwO_3Qa

+
(Tw_3QK
9(Alš2š2–
/
lšl2š
+
2}p;%8†%
/
qlq2q
+
;z8#•(AY43b;4);4;C!CT8
Bài toán 6:KA3;8TQqaqKZ>b;4);;8! ;;3!#`!;4

+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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•
;;€;4 YTY38
#4_;4`;4`8(P;88~Y’%";4%}m;:YA;
lR(SYAC;8U3(r\(t(P;8Y3U3;"(S
;T;#"wC;8!
JF.)y;
/ &%3`//
+ &8;R`//
> &
, a;8;@;4‘
E uKn:`;4`zvU3;$[C&
< u›U3n:`8(P;Y}vU3;#4]23!
I•
= uK(PrƒY}vU3;2€cDlR
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Ž  %• !•.
k:BY;]N:P5 q8dC•‘57†]‘6^:6•6’8
Tân Phú, ngày 18 tháng 05 năm 2014

)z%“„.
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