. suy ra s = ^ •. Vay
. VV=.$v3 V= r "
7
' n n i u q-•
j
kinh mat cau ngogi, npi ti§p tCp dipn
R
3
C h u n g minh: — >
. Khi nao d i n g thCrc xay ra.
2
A
•
T u o n g t y cho cac Sa, Sb, Sc roi cpng lai ta du-p-c:
. (9^^
343'''"
B a i t o a n 14. 3 0 : Cho tu- d i ^ n vu6ng OABC dinh O. Gpi R, r l l n lu-p't la
r
Trong tam giac A B C ta c6: a^ + b^ + c^ > 4 S
2(a' + b ' + c ' + a ' ' + b'" + c'') > 4 Vs .S,p. Do do SR^ >
^3
'
ban!
Dau bang xay ra khi tu- di^n A B C D d i u
*
^
X6t phep vj t y tam G ti k = j
_
thi tCf dipn A B C D b i l n thanh tu- di$n c6 4 dinh
1^ 4 trpng tam A'B'C'D' cua 4 m^t va R = 3R'.
Dgt OA = a, O B = b, O C = c
Vi R' > r => R > 3r => dpcm.
Bai t o a n 14. 3 2 : Cho \\Je di^n A B C D c6 cdc du-dng cao AA', BB', C C , DD' dong
R=iVa^+b2+c^
2
quy tai rnpt d i l m H thupc mi§n trong cua tie di$n. Cac du'6'ng t h i n g AA',
BB', C C . DD' lai c i t mat cau ngogi ti6p tLc dipn A B C D theo thu- tu' tai Ai, Bi,
1
abc
^
3V ^
^ 3 V 3 V.
Hu'O'ng d i n giai
Ta CO r =
Hhang Vi^t
^
- ( a b + bc + ca) + - \ / a V 7 b V T c V
2
r.
^ u -
AA'
u
Ci, Di. Chu-ng minh:
^'
2
^
BB'
+
AA,
CC
+
DD'
+
BB,
CC,
8
>-.
DD, 3
A
Hu'd'ng d i n g i a i
R
a b + be + c a + V a V + b V + c
V
TCP di0n A B C D dS la tu- di^n t r y c tam nen
b^+c^
2abc
A' la tru-c t a m t a m giac BCD. Gpi J la
4.4
abc
giao d i l m cua BI v a i mat cau ngogi ti§p
4
3N/3 + 3
tu' di$n A B C D thi A'l = IJ.
2abc
r
Do H la t r y c tam tam giac ABI nen:
D i n g thLPC xay ra khi a = b = c.
A'H.A'A = A'B.A'I = - A'B.A'J = - A'Ai.A'A •
2
Bai t o a n 14. 3 1 : Cho r, R l l n lu-pt la ban kinh m§t cau npi tidp, ngogi tiep cua m
=>A'H= -A'.Ai
2
tCp di$n CO t h ^ tich la V. Chu-ng minh rSng: 8R^r > 3 Vs V. Suy ra V <
Hu'O'ng d i n g i a i
Tu-ong t y : B'H = - B'B,; C H = - C C i , D'H
2
2
<•
Gpi O, G l l n lu'p't IS tarn mSt cSu ngoai ti4p vS trpng tSm tup di$n ABCD GPi
BC = a'. A D = a', CA = b', B D = b', A B = c, C D = c'. Gpi Sa, Sb, Sc, Sd, S,p
VHBCD ••• VHCDA + VHDAB + VHABC = V
'ABCD
lu'p't Id dien tich cdc m$t d6i d i ^ n v&\c dinh A, B, C, D vS d i ^ n tich ^o3<^
phSn cua tCr d i ^ n .
A B ^ = (OB - 6Af
i
= 2R2 - 2 0 A . 0 B ^
2 0 A . 0 B = 2R^ - A B ^
Mat khac 4 0 G = O A + O B + O C + O D
V,
=:> _JjBCD ^ " H C D A
V
V
ABCD
I
HA'
HB' H C
^
~r-~-i
+
AA'
^
a ' + b ' + c ' + a-' + b'' + c'' < 1 6 R '
^
.
^ v^HDAB
, V
" H. A B C ^ _ .j
Y/VBCD
''ABCD
^
=> 1 6 0 G ^ = 4R^ + S(2R^ - A B ^ ) , v6'i S la t6ng theo 6 canh
= 1 6 R ' - ( a ' + b ' + c ' + a-' + b'' + C ' ) > 0
2
BB'
HD' ,
+
CC
=1
DD'
AA,
BB,
CC, DD,
AA'
BB'
CC
DD'
YftBCD
A'A, B'B, C C
1+—Hj
AA
BB'
1+
CC
D'D
1-2
DD'~ M
^ A..
'
W trgng diem bSi dUOng hqcsinh
gl6i mon
Toan
1£
Lc Hoanh
L i t / iivnn mi v uvvn
'^hd
Theo bit ding thCpc B C S :
AA'
BB'
•
_AA,
+
—
BB,
CC'
+
DD'
+
CC,
DD;J
AA' BB' C C
AA, ^ BB, ^ CC,
AA'
BB'
CC'
DO-
NAhj^r)^
>16
DP' ^ 8
DD, " 3
A"i,A'2,A'3,A'4.Chu'ngminh:
i=1
; XGA,6.
i=i
n2
Dod6: i G A , < - l ZGA,
i=i
4
-14
4
4
(ab + bc + ca)
-fr.
•- ,-
r,i,r
Sxq(A'.AB'D')
—
1
..
Gpi G Id trpng tdm tip dien GA + GB + GC + GD = 0
3
3
3
1
vd GA = - ma, GB = - mt, GC = - mc, GD = - mj
4
4
4
4
Ta CO : AR^ = OA^ + OB^ + OC^ + OD^
Theo bat ding thu-c BCS :
y—^-XGAfi—
ttGA,
4tt
'ttGA,
IGA, <(R2 - 0 G 2 ) i - l - (dpcm).
i=i
i=i ^ ' ^ i
Bai toan 14. 34: Cho hinh hpp chu- nhgt ABCD.A'B'C'D'. Gpi R, r, h, V Ian lu^'
Id ban kinh mdt cdu ngogi tiep, npi tiep, ducyng cao ke tu- A' vd the tich cua
di$n A'AB'D'.Chung minh;
f
=>4R^>A(,^^+m^m^m,^)
«
tLK
1
—
3R'
= 40G2 + GA2 + GB2 + GC2 + G D ' + 20G(GA + GB + GC + GD)
= 40G^ + GA^ + GB^ + GC^ + GD^
GA^ + GB^ + GC^ + GD^ < 4R^
i=i
Vd 4XGAf > IGA,
3V
^(AB'D') _ Sxq(A.A'B'D')
2 ab + bc + ca 2
V(h-r) 2
= —.
< — => —^
<—
3R2
3 a^+b^+c^ 3
R2j.h ~ 3 '
Bai toan 14. 35: Ti> dien ABCD npi tiep trong mdt clu (O, R). Gpi ma, nib, nic,
md la dp ddi cdc trpng tuyen ve tu- A, B, C, D.
3
Chu-ng minh R > — (ma + mt + mc+ md)
16
Himng din giai
Suy ' ra
BDT« t G ^ < ( R ^ - 0 G f t - l i=1
i=i '^'^i
GAf\ = /->A2
OAf + OG' + 20AiOG= R^ + OG' + 20G(GA. - GO)
n2
,2 3.V
R2.
_ ^tp
Ti> dien A'AB'D' vuong tai A' nen R = V a ^ T b ^ T c ^
^
^ _ 1 _ < ^ ^ .
Hifang din giai
Gpi O va R Id tdm vd bdn kinh cua m$t c^u (S).
Ta c6: GAiGA', = R^ - OG^
i=i
^tp
=
vi^
3.V
^(AB'D')
vo'i Sxq(A A'B'D')
Bai toan 14. 33: Cho ti> di^n A1A2A3A4 c6 G Id trpng tarn, gpi (S) Id m|t cly
ngoai ti§p tu- di^n tren. Cac du-ang thing GAi, GA2, GA3, GA4 cit (S) tai
'
3.V
V
nnang
mf + m^ + m^ + m^ > -1 (ma + mt + mc + md) ^ => (Jpcm.
^ai toan 14. 36: Cho tCf di^n OABC trpng d6 OA, OB, OC dpi mOt vu6ng g6c
vai nhau, c6 dirdng cap OH = h. Gpi r Id bdn kinh mdt clu npi ti6p tCr di$n.
Tim gid tril6n nhltcua - .
r
Himng din giai
OA = a, OB = b, OC = c.
X^^zll < i .
R2.r.h 3
Hipang din giai
D$t AA' = a; AB' = b; A'D' = c. Ta c6
Tac6:l. = -L.-1, vd r = 3V
h^
l\/Id
a'
tp
-^'P - ^AOAB + S^oBc + S^oc^
3V
3V
r
3V
+ S^gg
1 1 1 1
a
b
c
h
181
.
diem
TDtri,ina
^.
hfli
1
JUcmq
1
man JOOm^ -
hoc '^inh qiol
••- "nnnil
riiu
1
t
Do do
=- +- + r h 2a b c
r 1 ^1
(^ 1 1^
<3
Ma
1
h—
[a
b 0^
y ^b^
0
n6n
+ —
l f
^1
1
ta
b cj
3
y'
1 1 1 N/3
< — = > - + - + - <
a
b c
0
+
1
V2-1
0 _
—
h
y
Do ( I 6 - - - < — = ^ - < - ( 1 + V3) .V#y - < 1 + N/3..
r h
h
r h
r
ii.».t.
h
Vay gia th Ian n h i t cua -
1^ 1 + Vs khi OA = OB = 0 0 .
r
Bai toan 14. 37: Cho hinh ch6p tii- gi^c d&u, gpi R, r Ian lu'p't la ban kinh mst
cau ngogi tiep va mat cau npi ti4p cua hinh ch6p 66. Tim gi^ tn \6fn nhat cQa
tfs6-.
^ •
R
•
12'
a
„
2 + tan
2
4 tan a .
R
A„-2(t-t^)
Xet ham so y =
^
.
.
1 + t'
^j^^^,,,,,
^
Trong mpi tam gidc a, b, c, dipn tich S thi:a^ + b^ + c^ > 4 V3 S
2(AB^ + AC^ + A D ^ + BC^ + BD^ + CD^) > 4
V3 S,p
Gpi O, G l l n lu-gt la tam va trpng tam tu- di?n A B C D , ta c6:
+ ( O D - O B ) ' + ( O D - O C ) '
= 1 6 R ' - ( O A + O B + O C + O D ) ' = 1 6 R ' - 1 6 0 G ' < 1 6 R ' = 16,.
_8_
V3-
a
Diu ding thu-c xay ra khi va chi khi AB = BC = CD = AC =AD = BD vd O = G.
Do do ABCD la tLP di^n d^u.
n e n r = IH = g t a n 2
tan^^-tan^^
2 _,
4 + 2tan^a
Hu-ang d i n giai
= (OB - O A ) ' + (OC - O A ) ' + ( O D - O A ) ' + (OC - OB)^
4 tan a
,^
gai toan 14. 38: Trong cdc tii- dipn npi tiep hinh cku c6 ban kinh R = 1 , tim tip
di^n CO dien tich toan phin Ian nhlt.
Do do S,p <
I la C h a n d u - a n g p h a n g i a c c u a g 6 c S M H
a
t a nI—
—
= V 2 - 1 khi a = 2arctan V V l ^ .
A B ' + AC^ + A D ^ + BC^ + BD^ + C D ^
4h
Do h = - t a n a => R = a.
Do do; - =
#
/ / 11
a^+2h^
a
R2 = (h - R ) ' +
R
Ap dung l l n lu'at vao cac mSt tCp di^n A B C D r6i cpng lai thi du-ac:
HiPO'ng d i n giai
Xet hinh ch6p ILF gi^c deu S.ABCD c6
canh day a, duang cao h. Gpi a Id g6c
hp'p bai mat ben vai day. Gpi O, I l^n
* luat tarn mat c l u ngogi tiep vd nOi ti§p
cua hinh chop thi O, I e SH.
Ta c6: OS.^ = OB^ = OH^ + BH^
^
V^y max
1 + tan^
2
v6it = tan^2 -^^(0:'')
2 ( - t 2 - 2 t + 1)
^ai toan 14. 39: TIP di$n ABCD c6 cdc cgnh AB, BC, CA d4u nho han DA,
DB, DC. Tim gia trj Ian nhat vd nho nhat cua PD, trong do P la d i l m thoa
(Ji4u ki^n PD^ = PA^+ PB^ + PC^
HiPO'ng din giai
Qpi O la diem sao cho O A + OB + OC - OD = 6
«
"•"a CO PD^ = PA^+ PB^ + PC^
^ ( 0 A - 0 P ) 2 +(OB-OP)^ + ( 0 C - 0 P ) 2 -(OD-OP)^ - 0
-
^20P2 - 2 0 P ( 0 A + 0 B + 0 C - 0 D ) = 0D2 - ( O A ^ + 0 6 2 + 0 0 ^ )
^ 20p2 = OD^ - (OA^ + OB^ + OC^)
y" = 0t = - 1 ± V 2 , c h p n t = V 2 - I .
(1)
(2)
^'fih phuang 2 ve cua (1), ta suy ra 20D^ - 2(0A^ + OB^ + OC^)
* DA^ + D B ^ + DC^ - (AB^ + BC^ + CA^)
(3)
4. '-t.,'"
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