1、相似三角形题型归纳总结非常全面相似三角形题型归纳一、比例的性质:比例的性质示例剖析(1)基本性质: = / 3x2(4)合比性质:- = - = (W*0) b a b aa 2 x+y 2+3“ _ 。 _ (v*0)y 3 y 3(5)分比性质:a-c Uh-Cl/ (加h0) b d b dy_3oy x_3 2(2 0)x 2 X 2合分比性质:b d a-b c_d(bd 工 0, a H b, c 工 d)x 2 x + y 2 + 3 z 八 一=厅0 (yHOn)y 3 x-y 2-3(7)等比性质:a c m z. f c、 = = =(b + d + “ h0) b d n
2、a+c + - + m a , Axzz =(Z? + d + + n H 0)b + d +打 b2 3 4已知-=-=- 则当x + f + zhO时,x y z2_3_4_2 + 3 + 4x y z x+y+z二、成比例线段的概念:1.比例的项:在比例式cr.b = c:d (即纟=上)中,a, d称为比例外项,b, c称为比例内项.特别地, h d在比例式ab = b.c (即上=?)中,b称为a, c的比例中项,满足b2=acb c2.成比例线段:四条线段6 b, G d中,如果Q和b的比等于C和d的比,即- = 那么这四条线b d段a, b, c, d叫做成比例线段,简称比例线段
3、.3.黄金分割:如图,若线段M上一点C,把线段朋分成两条线段AC和BC (AC BC),且使AC是和BC的比例中项(即AC2 =AB BC ),则称线段AB被点C黄金分割,点C叫线段 &8 的黄金分割点,其中 AC = 1ABQ.61SAB , = Q0.382AB, AC AB2 2的比叫做黄金比.(注意:对于线段A3而言,黄金分割点有两个.) A C B三.平行线分线段成比例定理1.平行线分线段成比例定理两条直线被三条平行线所截.所得的对应线段成比例.简称为平行线分线段成比例立AB DEBC EF如AF BEAC ABAE _AF AE _AF EBFC ABAC=SL EFT/BC &
4、FABCsMBC ZB = ZB, ZC = ZCrZA = ZAAB _ BC _ ACA = WC = ACAA【小结】若将所截出的小线段位置靠上的(如&B)称为上,位置靠下的称为下,两条线段合成的线段称为全,则可以形象的表示为二=二,空=刍 r r 全全2.平行线分线段成比例定理的推论平行于三角形一边的直线,截其它两边(或两边的延长线),所得的对应线段成比例.如AE AF AEEF = AABC ABC AM、AH AD AABC BC A!Mf A!Hr A!D9 AA0C BfCAB _ BC _ AC AM _ AH _ AD 7 = C = AC= =Ar = A7T=WD;AA
5、BC /A!BfCAB BC AC AB + BC + AC ;而一而一而一 A + BC + AC 一EB FC AB B C B、 C9 = Z4 ZZ? = ZZT AABC s MBC砂 BC ACSCsMBCAB ACA AC ZA_ZAABCs/WBC4DE / BC oHADE sAABC o A - AE - DEAB AC BCA BAAB CD oAOB s HCOD O 竺=竺=竺CD OC ODDG _ ANABC AADGsABC BC ZBAC = 90 /ADGsHEBDs&GCsMBCEM F CAEAAAAZAED = ZBAABCszMEDAE AC = A
6、D AB/IcAEAD _ACDEBCBAZACD = ZBABCsAACD/dACADCD八/AC2 =AD AB)/ABACBCBcA/ZAED = ZBABCAAED/AEADDEAE AC = AD AB、 厂ABACBC aAB1.BDAABC sMDEAB DE = BC CDED 丄 BDAC 丄 ECBDAABC s*DE s AACEEZABC = ZCDE = ZACEZABC sMDEAB DE = BC CDAB BC ACCDDECEC BDAABC s*DE s AACEAD ACMBCMCDE & =忑 C BD BJCDAB ACAB BCZABC = ZACE
7、AABC ZA4CAB BDAC = CDAAABD ACADZB = ZCADZC = ZBADAB2 =AD2+BD2 AC2 = AD2 + CD2 BC2 =AB2 + AC2D CC CE/AD BA E CE/AD Z1 = Z Z2 = Z3 AD ZBAC Z1 = Z2AE = ACCEAD =竺竺=竺AE CD AC CDAABC ABACAB BDAC = CDAF ;V EAw/nB M CBMCEN BM.EN BMEF / BCEF / BC 一NF MCNF MCx + 3y zx-3y+ zabc H 0a + b . x = k y =3k z=5kx + 3
8、y - z k + 9k-5k5=c_2bx 一 3y + zk-9k 5k311 -2x: y = 2:3x + y5y-x 1x _ 1x + 13y3y 32y 3y + ix: y: z = 1:3:5- V + 加=动=4 x y z 3x- yD2a-c + 3e b + c-ac + a-bx 1 工_ a + b c (a + b)(b + c)(a + c) y 2b d + 3/ abc abc4“ _ c _ 幺 _ 2 a+ cK77 3 b + d2a c + 3e _ 22b-d + 3f 3b + c - a c + a-b a + h-c (b + c- a)
9、+ (c + a-b) + (u + h c) = = = =1 a b c a+h+c(a + h)(h + c)(a + c) =8abcubc= =丄 n dfabc 7从矿百 Qbecf 44b + c = 2a, a + c = 2b, u 十 b = 2ca+h+c=O(a + b)(b + c)(a + c) (c)(一“) (-b)DE =EF =ABE121 / 厂U FAC4E AD = 8 DC = 6 AC = 10 .123x 6 = 2g5 2 sg沁善忌CE弓BC弓ABC ZXDEF ZA = 90 =90。AC = 5 BC = 3 DF = 0 EF = 26
10、 ZC = 85 ZE = 85=AB= AC = 15 BC = 2 EF = 8 DE = 10 FD = 6 ZA = 46 4 = 80。Z = 45 BC DF AD = AC ZFDC = ZACB DE EB = EC ZABC = ZFCD AABCsLCD (3)由等腰直角三角形得到心加Mac条件变为妙冷倍巴题型一亀 财:字和“8”字模型例题1 (1)如图4-1,已知口A3CD中,过点8的直线顺次与AC. AD及CD的延长线相交于点Q F、G,若BE = 5, EF = 2,则FG的长为 解析:(2)如图42,已知在口4BCD中,M、N为的三等分点,DM、DN分别交AC于几Q
11、 两点, AAEF sMEB , AGFDAGBC, :. = =19 :. 2L=AI) AF =1CBEB5 CBCB5 FGDF 3,即FG3得FG = 105BGCB 5FG + 75(2) !3由DC AB ,得APPCAM 1= = 9AB 3= -AC ,4同理2AQ=AC ,5吟获冷心却C,心AC,故Ap:p0:ec = l:| = 5:3:12巩固4 (1)如图4在ZV1BC中,M、E把&C边三等分,MN/EF/BC. MN、EF把ZVIBC分成三部分,则自上而下部分的面积比为 .(2)如图42,AB、CD、EF都与BD垂直,垂足分别是B、D、F,且AB = 1 , CD = 3、则 的值为 .(3 )如图43 已知在平行四边形&BCD中.M为的中点,DM, D3分别交&C于P, Q两点,则AP:PQ:QC = 图44图43解析:(1)1:3:5: (2
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