2026初中数学9年级上(湘教)专题3.2 平行线分线段成比例(讲义)(原卷版)

专题3.2平行线分线段成比例（举一反三讲义）





1

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2“#”

...................................................................................................................................................3



3“X”

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4“A”

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5“8”

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6!"#$%&'()*(+,

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7!"#$%&'(-.(+,

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8/012!"#34

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95678!"#

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10578!"#

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知识点平行线分线段成比例

1.9:;<

=>?@ABCDECF(GH"#I

2.J

KLMN

OPEQR?SSE!TU?mEnVWAEBECEDEEEFEX=E=

퐴퐵퐷퐸퐴퐵

푙1푙2푙3퐵퐶퐸퐹퐴퐶

E=EI

퐷퐸퐵퐶퐸퐹

퐷퐹퐴퐶퐷퐹⋯

3.YZ

V%&'A[(?D\]=[(^=[_`)CF(GH"#I

OPa1bcPa2bCdESEXe=E=EI

퐴퐵퐴퐶퐴퐵퐴퐶

퐵퐶퐷퐸퐵퐷퐶퐸퐴퐷퐴퐸⋯

【题型1判断比例式正误】

1

a24-25fghi·jklmno·p"qrbOPEWDcEcF!Ts([cctE

uvccEUVWEw['xw['EXiyzA{|(}a~~b△퐴퐵퐶퐴퐵퐴퐶퐵퐶

퐷퐸퐸퐹퐴퐹퐴퐹퐷퐸퐺퐵퐹퐸퐷

AI=BI=CI=DI=

퐴퐷퐷퐸퐴퐸퐵퐶퐴퐷퐷퐺퐴퐺퐷퐸

퐵퐷퐴퐶퐴퐸퐹퐺

1-1

a퐶퐹24-25fght·€‚ƒ퐷퐸·„…bOPE퐸퐺$†UVW퐵퐶GE‡

===3EXiyˆZ)‰(}ab퐴퐵∥퐶퐷∥퐸퐹,퐴퐹퐵퐸

퐴퐺1,퐺퐷2,퐷퐹

1111

IIII

A=2B=5C=2D=5

퐵퐺퐵퐺퐵퐶퐴퐵

퐺퐶퐵퐸퐵퐸퐸퐹

1-2

a24-25fght·lŠ‹·„…bOPEs)EWDstEWEEFstEŒ

EEXiyˆZ)E‰(}ab△퐴퐵퐶퐴퐵퐴퐶

퐷퐸∥퐵퐶퐷퐹∥퐵퐸

AI=BI=CI=DI=

퐴퐹퐴퐸퐷퐸퐷퐹퐴퐷퐹퐸퐴퐹퐷퐸

퐴퐸퐵퐶퐴퐵퐴퐷

1-3

OPEs퐴퐶)E퐵퐸c!Txc[t(W퐸퐶EWx퐵퐸[tAWEuvU

VWIXiyˆZ)A{|(}a△퐴퐵퐶퐷퐸b퐴퐵퐴퐶퐷퐸∥퐵퐶퐹퐵퐶퐴퐹퐷퐸

퐺

AI=BI=CI=DI=

퐴퐷퐴퐸퐴퐺퐴퐸퐵퐷퐶퐸퐴퐺퐴퐶

퐴퐵퐸퐶퐺퐹퐵퐷퐴퐷퐴퐸퐺퐹퐸퐶

【题型2“#”字型】

2

a2025·lŽ·‘’“bOPEsw[')EEWEstEUVW

FE‡=1:3E=3EX(`xab퐴퐵퐶퐷퐴퐷∥퐵퐶퐴퐵퐸퐹∥퐴퐷퐶퐷

퐴퐸:퐴퐵퐷퐹퐹퐶

AI6BI4CI5DI4.5

2-1

a24-25”ghi·•–`—·„…bOPEE‡=2EXxab

퐴퐵퐷퐸

퐴퐷∥퐵퐸∥퐶퐹퐵퐶퐷퐹

23

AI3BI2CI2DI3

2-2

OPE?abcEWAEBs?atEWCEDs?ctE"ACEBD!TU?b

VWEEFEXiy"($∥∥A{†˜(}ab

퐴퐸

퐴퐶

AIBICIDI

퐶퐸퐵퐹퐵퐹퐴퐵

퐴퐶퐵퐷퐹퐷퐶퐷

2-3

a24-25fght·lŠ™·p"qrbOPE?E‡=3E=4EX(š

퐵퐷

}ab퐴퐵∥퐶퐷∥퐸퐹퐴퐶퐶퐸퐵퐹

4433

AI7BI3CI7DI4

【题型3“X”字型】

3

a24-25”ghi·•–`—·„…bOPEE!TUccVWccE!TUcc

VWccE‡=6E=4E=3EX푎"∥푏∥푐(`xa푚푎b푏푐퐴퐵퐶푛푎푏푐

퐷퐸퐹퐴퐵퐵퐶퐸퐹퐷퐸

AI1.5BI4.5CI7.5DI10.5

3-1

a24-25”ghi·-›·„…bOPEE$†UVWE‡==3E

퐴퐵∥퐶퐷∥퐸퐹퐴퐹퐵퐸퐺퐴퐺퐷퐹

=1EX(šxab

퐵퐶

퐺퐷퐵퐸

3234

AI4BI3CI7DI7

3-2

a24-25fght·€‚ƒ·p"qrbOPEQR?E?!T$?cc

13123

UVAcBcC%WE?!T$?ccUVDcEcF%WE푙∥푙2∥$푙UVW퐴퐶OE‡=푙푙E푙

=3EX(`}퐷퐹I푙1푙2푙3퐴퐶퐷퐹퐵퐶퐴퐵

퐷퐸퐷퐹

3-3

QR%>Ϡ(?EE!TD?l4VWEEED?VWEEE?

$†UVWOEŒ=32E=푙150푙2E푙3=8E=2I퐴퐵퐶푙5퐷퐸퐹푙4

푙5퐴퐵퐵퐶퐸퐹퐸푂

(1)3(`

(2)3퐷퐸(`I

푂퐵

【题型4“A”字型】

4

a2025·€ž·‘’“bOPEŸ[

¡¢F£E‡==6E

=4EX(`xab△퐴퐵퐶퐵퐶△퐷퐸퐹퐸퐶2퐵퐸

퐶퐺퐷퐹

AI6BI8CI9DI10

4-1

a24-25”ghi·¤¥¦§·„…bOPEs)EE!T}[Et(WEŒ

I‡=2E=4E=1EX(`}ab△퐴퐵퐶퐸퐹퐴퐵퐴퐶퐸퐹∥

퐵퐶퐴퐸퐵퐸퐴퐹퐹퐶

1

AI2BI1CI2DI3

4-2

a2025·€·)¨©

bOPCd(ª«)E¬­®¯'([`°x1E(%­±W°

sª«(UWtEWDcE!T}[c$ª«(UWEuvEX(`xa△퐴퐵퐶b

퐵퐴퐶퐴퐷퐸퐷퐸

1

AI2BI1CI2DI3

4-3

a24-25”ghi·¤¥¦§·„…bOPEs3×4(¯'ª«)Eccx«WEuvE

퐴퐵퐶퐶퐷

U²W(³«VWI‡®¯'[`x1EX=I

퐴퐸퐶퐸

【题型5“8”字型】

5

a2025·´µ¶·%bOPE)EExG&tAWE²WE(?!TU[E

VWFEGEU·EVWME▱퐴퐵퐶퐷NI‡=3E=2EX퐵퐷(šxab푀푁퐴퐵

퐵퐶퐷퐴퐷퐶푀퐹퐸퐹퐸퐺⋅퐸푁

AI6BI8CI10DI15

5-1

OPEsw[')E(!BF!T$ACcADUVWEcFE=3E

퐴퐵퐶퐷∠퐴퐵퐶퐴퐵

=2EX(šxab

퐸퐹

퐹퐷퐹퐵

2333

IIII

A5B8C7D5

5-2

OPEABECD†UVWEEŒACEFDBEWCEFEBs¸A>?tEQRAC¹pEEF¹

rEDB¹qEXpEqErº»¼½(J¾¿}a~~b∥∥

1111

AI+1=1BI+1=2CI+1=1DI+1=2

푟푞푝푝푞푟푝푞푟푞푟푝

5-3

OPEsw[')EEx()WEFxtAWE$UVWHE=3cmE

=6cmE=4cmEX(`xa~~b퐴퐵퐶퐷퐴퐵퐴퐷퐸퐹퐴퐶퐹퐻

퐸퐻퐴퐻퐻퐶

AI22cmBI20cmCI18cmDI16cm

【题型6平行线分线段成比例与三角形的中位线的综合】

6

a24-25fghi·ÀÁ·p"qrbOPEs˜ÃRt)E==}"tAWE

uv=15°EÄ"ÅWÆÇÈÉÊ90°F£Euv△퐴퐵퐶U퐴퐶VW퐵퐶EX3,퐷(`Ëx퐵퐶I

퐴퐷,∠퐵퐴퐷퐴퐷퐴퐴퐸퐵퐸퐴퐶퐹퐶퐹

6-1

a24-25”ghi·ÌÍ·„…bsw[')EG&E†UVWOE

I‡=4EX"(`xI퐴퐵퐶퐷퐴퐶퐵퐷

푂퐸∥퐶퐷푂퐸퐶퐷

6-2

a2025·ÎÏз‘’“bOPE}(&!EE}()WEuvUVW

FE‡E==8EX"(`x퐵퐷△퐴퐵퐶I퐴퐵퐶퐸퐵퐷

퐶퐸⊥퐵퐷퐶퐸2퐵퐷퐴퐵

6-3

OPEDEcNM!T}ABCcADE()*ENM(_`UBCVWFEX

Sw['

△퐷푀푁

MFCE˜Vab△△푆

AI1

5BI1

4CI2

5DI2

7

【题型7平行线分线段成比例与三角形的重心的综合】

7

OPE}(-.E_`UVWE_`UVWEE!T}Ñ

(-.E`x퐺12EX△퐴퐵퐶(`xab퐵퐺퐴퐶퐷퐶퐺퐴퐵퐸푃푄△퐵퐶퐸△퐵퐶퐷

퐵퐶푃푄

AI2BI2I5CI3DI4

7-1

OPEWx(-.E=90°E=30°EuvÒ_`UVWE5V

퐺△퐴퐵퐶∠퐶∠퐵퐶퐺퐴퐵퐷퐷퐸⊥퐶퐵

WE²W5UVWEX(šxa~~b

퐺퐹

퐸퐺퐺퐹∥퐴퐷퐴퐶퐹퐷퐸

4323

AI1BI3CI2DI3

7-2

OPEs)E}[t()EWG}(-.E²WG5UVW

△퐴퐵퐶퐴퐸퐵퐶△퐴퐵퐶퐺퐹∥퐴퐵퐵퐶

FEÓÔ=ab

퐸퐹

퐸퐶

1111

AI3BI2CI4DI5

7-3

OPEsÕ')E=6EP}[tAWEğÖ×E‡WD(GHWE

ØÙ}(-.EX퐴퐵퐶퐷(`x퐴퐵I퐴퐷△푃퐶퐷퐶푃

△퐴퐵퐶푃퐷

【题型8多次利用平行线分线段成比例求解】

8

a2025fght·ÚÛ·Ü

qrbOPEQREE}%­Ú˜(˜Ã%&'E

Ý[EEs¸A>?tEŒ=3E=△3E퐴퐵퐶!TU△퐷퐶퐸E△퐹퐸퐺EVWEEI4Þi

yß

퐵퐶퐶퐸퐸퐺퐴퐵퐵퐶퐵퐹퐴퐶퐷퐶퐷퐸푀푁퐻

(1)(šx__________

퐹퐺

퐵퐺

3

(2)3à

==3

퐹퐺퐸퐺

(3)3

퐵퐺퐹퐺(šI

퐵푀:푀푁:푁퐹8-1

OPEsÑ)EDcEcF!Ts"cctEuvcEE

E=1=△3E3퐴퐵퐶(`I△퐴퐶퐺퐴퐵퐴퐶퐴퐺퐷퐸퐸퐹퐷퐸∥퐵퐶

3

퐴퐷

퐸퐹∥퐶퐺퐴퐵,퐴퐹퐴퐺

8-2

OPEWD}[tAWEuvE²tWE5EUVWFE²WF5

3

UVWEQR△퐴퐵퐶E퐵퐶I퐴퐷퐴퐷퐸퐹∥퐵퐷퐴퐵

BCG=2=4

퐴퐸

퐹퐺∥퐴퐶퐸퐷퐵퐺

(1)3(`

(2)‡퐶퐺=2EstN>áшZiE3(`I

퐶퐷8-3

a24-25”ghi·•–`—퐸퐹·„)b®â¸

s

r†ã%&'Çä£åæA­ß

OPEs

)EWD}()WEWE})WIuˆEUVWGE3(šI

퐴퐺

Δ퐴퐵퐶퐵퐶퐴퐶퐴퐷퐵퐸퐷퐺

a1b®âçèE²WD5UVHEéê!"#ëìF£ß

(ÞíIiî}®â

(ï!4

²ð

퐷퐻∥퐵퐸퐴퐶

4

OP1E²WD5UVHE

퐷퐻∥퐵퐸퐴퐶

}()WE

∵퐷퐵퐶

=E

∴퐵퐷퐶퐷

==1E

퐶퐷퐶퐻

ñòóÚôi(4

²ðI∴퐵퐷퐸퐻

õöH2

a2bOP2Es)EEcF!T}c()WEuv!TUcVMcNEX=_______I

푀푁

▱퐴퐵퐶퐷퐴퐵퐶퐷퐴퐶퐷퐸퐵퐹퐴퐶

÷øùú

a3bOP3EWE}[()WE=cUV=10E=6,E?vûüw['

(îýI퐴퐶퐷퐶2퐵퐷,퐴퐷퐵퐸퐹,퐴퐷퐵퐸퐴퐷⊥퐵퐸

퐶퐷퐹퐸

【题型9作垂线构造平行线分线段成比例】

9

a24-25fghi·¤¥þÿ·„…bOPEsÕ')E=3E=4E†UVWOEE

x_`tAWEuvUVWFE‡=1EX퐴퐵퐶퐷(`Ëxa퐴퐵b퐴퐷퐴퐶,퐵퐷

퐵퐴푂퐸퐴퐷퐴퐸푂퐸

4127265

AI2BI2CI2DI2

9-1

a2025·lĀā·%bOPEs)E}()E_`ĂWEă

=EuvE‡==45°E△퐴퐵퐶=62퐶퐷EX△(`xa퐴퐵퐶b퐶퐵퐸

퐵퐸3퐵퐶퐴퐸∠퐸퐴퐵∠퐵퐷퐶퐴퐵퐵퐶

52

AI23BI3CI2DI13

9-2

a2025·Ą¥ąĆ·ćbOP1EÄ[`x4(˜[Ÿ\[t(ú