Cho AABC co H la tnrc tam, 0 la tam dlti1ng tron nqoal tiep va M la trung diem BC.. Cho AABC nhon n¢i tiep duong tron tam O.. Chltng minh rang cac duong qua M song song vai OA, qua N son
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•
Tinh chat sau day cUa trvc tam tam giae co
nhieu Lmg dl,lng trong giai cac bal tap hlnh hQC.
Tinh chat. Cho AABC co H la tnrc tam, 0 la
tam dlti1ng tron nqoal tiep va M la trung diem BC.
The thl AH = 20M.
Chung minh. Xet tntong hop AABC nhon (cac
tntong hop khac ehLmg minh tuang tl/).
Va duong kinh BD ella (0),
VI DAB = 90° nan DA.l AB.
Ma CH .1 AB nan DA II CH.
Bili toan 1 Cho AABC nhon n¢i tiep duong
tron tam O. G9i M, N va Plan luqt la trung diem
cac canh BC, CA va AB. Chltng minh rang cac
duong qua M song song vai OA, qua N
song song vai OB va qua P song song vai OC
Loi giai. G9i H la true tam AABC; I va Elan
luqt la trung diem ella HA va HO.
ViAH =20M nan AI = IH=OM.
Ma AH 110M (do eung vuong g6e vai BC) nan
cac t(( giae AIMO va IHMO la hinh blnh hanh
MOT TINH CHAT
D';NG HAl GIANG
(GV THCS Thl tran Xuy€m, HiJ Tinh)
Suy ra MI II AO va E la trung diem ella MI. T((e la E thuoc duong qua M song song vai OA.
Tuang tl,l E cling thuoc cac duong qua I\J
song song vai OB va qua P song song vai OC. Tlt d6 suy ra dpem.
Bid toan 2 Cho AABC nhon e6 H la tnrc tam. Chltng minh rang 9 diem g6m chan ba duong cao, trung diem ba canh va trung diem cac doan
HA, HB, HC dong viano LOi giai. sadunq hinh ve va ehltng minh ella
ED =EM =EI = MI = OA.
Sa dl,lng cac ket qua tuang tl,l va ket hop vai
OA = OB = OC ta suy ra 9 diem da eho cling thuoc duong tron tam E.
Bili toan 3. Cho AABC nhon n¢i tiep duong tron tam 0 co cac duong cao·AN va CK. Duong tron nqoal tiep aBKN cat duong tron (0) tal diem th(( hai M. G9i I la trung diem ella AC. Chltng minh 1M .1 MB.
LOi giai. G9i H la tnrc, tam ella AABC, J la trung diem BH.
Ta thay J la tam duong tron tiep aBKN
va BH la duong kinh ella (J).
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VI IHJO la hlnh blnh hanh nan IH II OJ.
khac, VI M, B la giao ella hai duong
tron (0), (J) nan OJ 1-MB.
Suy ra IH 1-MB.
Ma BH la duong kinh ella (J) nan HM 1-MB.
Tli d6 suy ra \, H, M hang va 1M 1-MB.
Bcii toan 4. Cho tCt giae ABCD n¢i tiep duong
tron (0 ; R). GQi H1, H2 H3, H4 thCt nr la tnrc tam
ella cac tam giae ACD, BCD, ABD va ABC.
ChUng minh rang:
a) BH1,AH2 CH3 va DH4 dong quy.
b) Bon H1, H2 H3, H4 dong viano
D
Li:fi giai a) Vi AH 1= BH2 (= 20M) va AH1II BH2
(vl cling vuong g6e vai CD) nan tCt AH 1H2B
la hlnh blnh hanh,
Suy ra AH
2 va BH1 cat nhau tai trung I ella moi duong.
ChUng minh tuang nr vai cac cap khac ta
suy ra BH 1,AH2 CH3 va DH4 dong quy tal trung
Bid toan 5. Cho dLtOng tron tam 0 ban kinh R :
va P c6dinh narn ngoai dl.tOng tron Va M'p : tuyen PA va cat tuyen PBC (A, B, C nam trEm (0)). :
Chang minh rang khi cat tuyen PBC thay d6i thi :
la trung ella AH. Lay 0' doi xUng vai 0 qua ·
Ta thay 0' la cO djnh va tCt giae AOPO' la · ·
Suy ra O'P II AO va O'P =AO. · ·
COng vi tlt giae AOIK la hlnh blnh hanh nan · ·
Suy ra O'P II KI va O'P = KI. Tli d6 tlt giae ·· ·
O'PIK la hlnh blnh hanh, Suy ra O'K II PI. ·
Ma PI 1-AK nan O'K 1-AK. ·· Suy ra can tal 0' O'H =O'A. · Tli d6 H thu¢e duong tron cO djnh tam 0' ban · ·
·
Bcii 1. ChUng minh rang tnrc tam, tronq tam va · ·
tam duong tron nqoal tiep ella m¢t tam giae cling · ·
. nam tren m¢t duong Bcii 2. Cho duong tron (0) va ba (duong A, B, C Ole). ·· · ·· thay 95i tren (0) sao eho tnrc tam H ella MBC la · ·
m¢t co dinh nam trong (0). Tim quy tieh · ·
b) Lay 0 1 doi xUng vai 0 qua I. chan cac duong cao ella MBC. ·
Trang 18THAI HOU (Gv THCS Quang U:,c, Can Lt?c, Ha Tfnh)
Co rat nhieu phLlong phap giai mt?t bet toen bat dlmg thLic (BDT), nhLlng chUng ta can tim mt?tphLlong phap giai 99n nhat chUng ta can t<;lokinang chLing minh BDT Khi tim ve BDT, chUng toi thay mt?t s6 BDT co bien s6xec d;nh trong esc doen thi ta can xem xet kTgia tr; cec bien s6 do ra mQt BDT dung, roi
ta BDT dung nay, ta bien d6i thanh BDT phai chLing minh
Bid toan 1 Cha cac so thuc x, y, Z E [1 ; 2]
thea man dieu x + y + Z = O
ChUng minh rang x2 + y2 + z2 s 6.
Loi gild VI x E [1 ; 2] nen (x + 1)(x 2)s0
<=> x2 ::; x+ 2
TL/dngnry2sy + 2; z2 sz + 2.
C¢ng ve thea ve cua ba BDT tren ta dL/<;lc
x2 + y2 + z2 ::; X+Y +Z + 6 =6.
thltc xay ra khi va chi khi x, y, Z E {1 ; 2}
vax + y + Z = 0 <=> trang ba so x,y,Zcohai 56
bang 1, so con lal bang 2
Bid toan 2 Cha cac so thirc x, y, Z E [0 ; 2]
thea man dieu x + y + Z = 3.
ChUng minh rang x2 + y2 + z2 s 5
Loi giai VI x, y,Z E [0 ; 2] nen
<=> trong ba so x, y, z co mot so bang '2, mot so
bang 0, so con lal bang 1.
Bid toan 3 Cha cac so th,!c a, b, C E [2 ; 5]
thea man dieu kien a + 2b + 3c ::; 2.
b2(1TLJOng t'! b(1 c) c); c(1 c2(1 a). C¢ng thea ve ba BDT tren ta suy ra
2a)
<=> 1 + (a2b + b2c + c abc (a2 + b2 + c2) + (1 a)(1 b)(1 c).
Ma(1 a)(1 b)(1 c) 0 va abc 0 nen suy
Loi gild Voi cac 56 th,!c x, y bat kJ ta cO
(x y)2 0 <=> (x + y)2 4xy.
Ap dl:mg voi x = 1, y =a1+ a2+ + a2010 ta cO (1 + a1+ a2+ + a2010)2 4(a1+ a2+ + a2010).
Trang 19.:xL Co ehinh phttong hhong?
Cho so t'! nhien S c6 2011 chlt so, trong d6 c6 2010 chlt so 5 va m¢t chlt so
a khac 5. Hei S c6 phai la so chfnh phuong hay khOng?
CAO QUaC CUONG (GV THCS Vinh Tuong, Vinh Pnuc)
Ox, Oy tuong ltng tal cac diem A 1' A2.
Tren (0) dlfng lien tiep cac diem cac diem A3
A4, ·· , A20 phan bi$t, khac A 1,A2 thea man A2A3
man dieu ki$n bal toan.
Cht1ng minh Theo each dlfng ta c6
= =A190A20 =ApA2 = A20A3 =A30A4
. Ma 192 = 361 nen A10A20 = =
hoac
Cac ban sau dU<;1c thuong ki nay: Tran Phuong Nga, 9C, THCS Thanh Thuy, Phu ThQ;
Nguyen f)(fc Th9, 9C, THCS Phan B¢i chau, Tlt
Ky, Hai Duong; Nguyen f)ang Huy, 6A, THCS Hoang Xuan Han, Dltc ThQ, Hli TInh; oso H6ng Qulm, 80, THCS Cao Xuan Huy, Dien Chau, An; Nh6m ban Le ChI Hieu, f)ao Anh Tuyet, Nguyen f)inh Hien, 57B Quang Trung,
TP. Quang Ngai, Quang Ngai.
ANH COM PA
Ma vai moi i E {1 ; 2 ; ; 2010} ta c6 aj :::>: af Ch ' ung min . h" rang a b c
Tlt d6 suy ra dpcm.
B" 3 Ch 1 <aj <.1 ,.
thltc xay ra khi va chi khi trong 2010 so 31. 0 2010 2009' Val a 1, a2, · ,
da cho c6 2009 so 0, so con lal bang 1.
Blii tap a2000 va b1,b2, , b2010 ta cac so th,!c duong. Blii 1. Cho cac so thuc a, b, c E [0; n Chltng minh rang
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