Vi A thuQC duong tron duong kinh CD nen A la giao diem ciia duong thang d va duong tron duong kinh CD, suy ra toa dp ciia A la nghi^m ciia he:
x + y - 5 = 0
2 , . . 2 . . " ^ A( 4; l ) ( v i x A > 0 ) .
X + ( y - 5 r =32
Suy ra A C = 8 => AB = 2S AABC AC = 6.
Vi B thupc duong thang A D : x - 4 = 0 nen B(4; y) Tu A B - 6 = > ( y - l ) ^ = 3 6 = > y = - 5 , y = 7
Vi AB va A D cung huong nen ta c6 B(4;7). Vay phuong trinh BC: 3x - 4y +16 = 0.
di^ 1.5.13. Cho tam giac ABC c6 M ( 2 ; 0 ) , N ( - l ; - l ) , P ( - 2 ; 3 ) Ian lugt la
trung diem cac canh AB,BC,CA . T i m toa dp cac dinh A , B , C .
Phucntg i>hapgiai Toihi Hiiih hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Tat Thu Xgigidị
Do M , N , P la trung diem cac canh nen ta C O AP = M N hay suy ra
=-5
y^-3 = -\2
Tudo, suy ra B(l;4),C(3;-4). =>Ă-5;2).
Vidu L5J4. Trong mat phSng Oxy cho tam giac ABC c6 M(l;0),N(4;-3) Ian lupt la trung diem cua AB,AC; D(2;6) la chan duong cao ha tir A len BC. Tim tpa do cac dinh cua tam giac ABC.
gidị
Goi Ăa; b ) , suy ra DA = (a - 2; b - 6), M N = (3; -3).
Vi A D l M N = : i > D A . M N = 0 c : > a - 2 - b + 6 = 0<=>a-b = - 4 (1). Lay doi xiing diem A qua M , N ta c6: B ( 2 - a ; - b ) , C ( 8 - a ; - 6 - b ) Suy ra BD = (a;6 + b), CD = (a - 6;b +12)
Vi B, C, D thang hang nen ta c6: ~—- - ~ — a b + 6
Tu (1) va (2) ta suy ra a - - 5 ; b = - 1 . Vay A ( - 5 ; -1), B(7; 1), C(] 3; - 5 ) .
<r>a + b = - 6 (2).
Vidu 7.5.15.Trong mat phang Oxy cho tam giac ABC c6 M(-1;-1),N(0;2) Ian lupt la trung diem cua AB, AC va D ( l ; 0) la chan duong phan giac trong goc Ạ Tim toa do cac dinh cua tam giac.
Xffigidị
Goi Ăa; b) => B(-2 - a; -2 - b), C(-a; 4 - b). Suy ra B D = (a + 3;b + 2),CD = (a + l ; b - 4 )
Vi B, C, D thSng hang nen — = o 3 a - b + 7 = 0=> b = 3a+ 7 (1). a+1 b - 4
Mat khac D la chan duong phan giac trong goc A nen | A D , A B - | A D , A C J cos (AD, AB) = cos (AD, A C AD.AB AD.AC
AB AC
Ma A D = (1 - a; -b), AB = (-2a - 2; -2b - 2), AC = (-2a; 4 - 2b)
80
Cty TNHH MTV DWH Khang Vi^t
^ . ^ ^ ^ ( a - 1 ) ( a . l ) . b ( b . l ) ( a - l ) a . ( b - 2 ) b 7(a + l)2+(b + l)2 ^ . r + ( b - 2) 2
, . a ^ - l + (3a + 7)(3a + 8) a ^ - a + (3a + 5)(3a+ 7) Thay (1) vao (2) ta co: ^ ^ ^
V(a + l)2+(3a + 8)2 Va2+(3a + 5)2 2â+9a + l l 2â+7a + 7
V2â +10a + 13 V2â +6a + 5
<»(2â +9a + ll)2(2a2 +6a + 5)-(2a2 +7a + 7)^(23^ + 10a +13) = 0 o â + 6â + 12a + 8 - 0 o (a + 2)(â + 4a + 4) = 0 <=> a = -2,b = 1 . Vay Ă-2;1),B(0;-3),C(2;3).
Vidu 7.5.76.Trong mat phang Oxy, hay xac djnh toa do cac dinh cua tam giac ABC biet M(l;4), N(-l;3) la trung diem ciia BC, CA va H
true tam tam giac ABC.
3' 3 la
£gi gidị
Goi C ( x ; y ) = ^ B ( 2 - x ; 8 - y ) , A ( - 2 - x ; 6 - y )
Vi H ( - ; - - ) la true tam tam giac ABC nen
3 3 AH.BC = 0 CH.MN = 0 Ma A H = (7 — + x; + V 23 [3 3 ^ , B C - ( 2 x - 2 ; 2 y - 8 ) , CH = ( i - x ; - | - y ) , M N = (-2;-l) Nen {*)<:> ( | + x)(2x-2) + ( - ^ + y)(2y-8) = 0 (1) 2 ( x - i ) . y . | = 0 (2)
(2) o 2x + y +1 - 0 y = - 1 - 2 x (3) thay vao (1) ta dupe :
(7 ^ - + x (2x - 2) + f — + 2x1(10 + 4x) = 0 V 3 J O l S x ^ + 86x + 123 = 0 o x = -3;x = - — 15 • X = -3 =:> y = 5 ^ Ăl; 1), B(5; 3), C(-3; 5) • V 41 67 ^ 15 ^ 15 r i i . 2 3 ^ , B '71.53^ c { 41.76^ t l 5 ' l 5 ; , B . 15'15, 81
Phuomg phap giai Todn Jjinh hqc theo chuyen de- Nguyen Phu Khanh, Nguyen Tat Thu
Vi du 1.5.17. Trong mat phang tpa dp Oxy cho tam giac ABC c6
D(2;-l),E(2;2),F(-2;2) la chan duong cao ha tu A,B,C . Xac dinh toa do cac dinh cua tam giac ABC.
gidi
Gpi H(a;b) la true tam tam giac ABC.
Ta CO tu giac BDHF, CDHE, BCEF la cac tu
giac npi tiep nen suy ra
HDF = HBF; HDE = HCE; HBF = HCE => HDF = HDE =:> A H
la phan giac trong goc EDF. Q
Tuang tu, ta c6 BH la phan giac trong ciia goc DEF. Suy ra H la tam duong tron noi tiep tam giac DEF. Ta C O :
EH.EF EH.ED
EF ED
FH.FE FH.FD
EF FD
, giai he nay ta tim dupe a = l,b = 1 hay H ( l ; l ) .
Suyra HD = (l;-2) nen phuong trinh BC : x - 2y - 4 = 0 . HE = ( l ; l ) nen phuong trinh AC : x + y - 4 = 0
HF = (-3; l ) nen phuong trinh AB : 3x - y + 8 = 0 . Vi A = A B n A C = ^ A : 3x - y + 8 = 0
x + y - 4 = 0 Tuong tu, ta tim dupe B(-4;-4),C(4;0).
x = - l
y . 5 - •Ă-l;5)
Vidu ị5.15. Trong mat phang Oxy cho tam giac ABC npi tiep duong tron
(C) CO phuong trinh: (x - 1 ) ^ + ( y - l ) ^ =10. Diem M(0;2) la trung diem
canh BC va dien tich tam giac ABC bang 12. Tim tpa dp cac dinh ciia tam giac ABC.
JUffigidL
Duong tron (C) c6 tam 1(1; 1), suy ra M I = (1;-1).
Vi BC di qua M va vuong goc voi M I nen B C : x - y + 2 = 0. Tpa dp B,C la nghi^m ciia he:
( x - l ) ' + ( y - l ) ' - 1 0 x - y + 2 = 0 y = x + 2 x 2 = 4 x = 2,y = 4 x = -2,y = 0 82
Cty TNHH MTV DWH Khang Viet
Suyra B(2;4),C(-2;0) hoac B(-2;0),C(2;4) Gpi Ăa; b), suy ra (a -1)^ + (b -1)^ = 10 (1) ,
a - b + 2
Ta c6: d(A,BC) = 4i ,BC = 4V2=:>S^ABC = 2 a - b + 2
Nen taeo |a-b + 2| = 6<=>a=b + 4,a = b - 8. , a = b + 4 thay vao (1) ta c6:
(b + 3)^ + (b -1)^ = 10 b^ + 2b = 0 o b = 0,b = -2
, a = b - 8 thay vao (1) ta eo: (b - 9)^ + (b -1)^ = 10 v6 nghi^m. Vay Ă0;4) hoac Ă2;-2).
Vi du 1.5.19. Trong mat phang tpa dp Oxy, cho tam giac A B C eo dinh
Ă3;-7), true tam la H ( 3 ; - l ) , tam duong tron ngoai tiep la I(-2;0). Xac dinh tpa dp dinh C, biet C c6 hoanh dp duong.
JCgi gidị
Cdch 1:CQ\) la trung diem cua B C , D la diem doi xung voi A qua O .
Ta CO B H // C D , C H // B D nen tu giac B D C H
la hinh binh hanh nen M la trung diem H D Tir do suy rạ A H = - 2 M I : .M(-2;3) 0 = -2(-2 - x) 6 = -2(-y) <=>s x = -2 y = 3 Nen duong thang B C qua M c6 A H ( 0 ; 6 )
la vtpt CO phuong trinh la: y - 3 = 0.
Gpi C(a; 3), do lA = I C 5^ + (-7)^ = (a + 2)^ + (3)^
o â + 4a -61 = 0 <=> a = - 2 ± V65 => C(-2 + ^y65;3).
Cdch 2. Duong tron ngoai tiep tam giac A B C eo phuong trinh (x + 2)^ + y^=74.
Phuong trinh A H : x = 3, do B C 1 A H B C : y = m ( m ^ - 7 ) Tpa dp B, C la nghiem eiia phuong trinh:
(x + 2)2 + m 2 =74c^x^+4x + m 2 - 7 0 = 0 (*)
Vi (*) CO hai nghiem, trong do c6 it nhat mpt nghỉm duong nen m <V70
Phuvtig phdp gidi Todn Hinh hgc theo chuyen df - Nguyen Phu Khdnh, Nguyen Tai Thu
Khido: B(-2 - V74 - ;m), C(-2 + \/74 - ;m)
Vi BH 1 AC => A C B H = 0« . m 2 + 4 m- 2 1 = 0<=>m = 3 Vay C(-2 + ^/65;3).
Vi diJ. 1.5.20. Trong mat phang tpa dp Oxy, cho hai duong thang
dj : Vsx + y = 0 va d j : \/3x - y = 0. Gpi (T) la duong tron tiep xuc voi di tai