有關(guān)調和性質(zhì)的應用

2012年2月觀(guān)周刊第8期總568期.February 2012 M i有關(guān)調和性質(zhì)的應用°戶(hù)淑茹②趙臨龍(安康學(xué)院數學(xué)與應用數學(xué)研究所安康725000中圖分類(lèi)號:0185.1文獻標識碼:A文章編號:1008-925X(2012)02-0229-01摘要:利用調和分割的定義解決初等幾何問(wèn)題:如何做調和共軛點(diǎn)、證明一角的兩邊和它的內外交平分線(xiàn)成調和線(xiàn)束、一線(xiàn)段被它的中點(diǎn)和這直線(xiàn)上的無(wú)穿遠點(diǎn)所調和分割、以及如何用完全四點(diǎn)形的調和性質(zhì)作圖關(guān)鍵詞:調和分割中點(diǎn)無(wú)窮遠點(diǎn)完全四點(diǎn)形The application concerning concordance propertyHu Shuru Zhao LinlongAbstract: The definition made use of to temper partition solution the elementary grade is several what problem: How temper conju-gate point and prove the both sides of dime with its inside the diplomacy divide equally line to become to temper line to tie, front-linesegment drive its middle point and this straight line up of the everlasting far order in harmony with partition, and how use up the inharmony with property of whole squareses to make diagram.Keywords: In harmony with partition MiddleThe everlasting far orders Complete square調和分割的定義:設A,B,C,D為共線(xiàn)的四點(diǎn),我們把由于∠2=∠TOB,∠1=∠BOTAG:B犯定義為這四點(diǎn)按著(zhù)順序的交比,并以符號(ABCD)∴∠1=∠2,同理:∠3=∠4表達CAB, CD)=B當(AB,CD)=-1時(shí),即C點(diǎn)分割線(xiàn)(由三角形中一個(gè)角的內角和外交平分線(xiàn)和對邊的交點(diǎn),調和分割對點(diǎn)段AB的值和D點(diǎn)分割線(xiàn)段AB的值只差一個(gè)符號(因而一個(gè)是從中學(xué)幾何,我們知道三角形中一個(gè)角的內角和外角平分線(xiàn)內分點(diǎn),一個(gè)是外分點(diǎn)),這時(shí)說(shuō)C,D兩點(diǎn)調和分割線(xiàn)段AB或將對邊分成兩線(xiàn)段的比值都和兩鄰邊成比例(注意中學(xué)幾何講的對C與D于線(xiàn)段AB成調和共軛點(diǎn)偶是絕對量!),可見(jiàn)兩條平分角線(xiàn)和對邊的交點(diǎn),調和分割對邊但是,在調和分割中,兩對點(diǎn)的關(guān)系是完全對等的,即當C與例3一線(xiàn)段被它的中點(diǎn)和這直線(xiàn)上的無(wú)窮遠點(diǎn)所調和D調和分割線(xiàn)段AB時(shí),A與B也調和分割CD.若(AB,CD)=分割。1,便也有(CD,AB)=-1。證明:設C點(diǎn)為線(xiàn)段AB的中點(diǎn),D。為直線(xiàn)AB的無(wú)窮遠點(diǎn)例1試證一角的兩邊和它的內外交平分線(xiàn)成調和線(xiàn)束。證明:以下引用的角均系有向角,即則(AB,CD)=xABC(b,c),(b,d)=90-(c,b由于(ABD)=mD=1,因此(AB,CD=)=(ABC)=(a,a)=90°+(a,c)=90°-(b,c)由上題還可知,共線(xiàn)四點(diǎn)的交比,若第四點(diǎn)為無(wú)窮遠點(diǎn),那么(ab, cd)=sin(a, c)) sin(b,d)- sin[ 90"-(c,b)]一交比值等于前三點(diǎn)的簡(jiǎn)比值sin(a,d)·sin(b,c)sin[90°+(a,c)]例4調和點(diǎn)列作圖:已知A,B,C求第四點(diǎn)Dcos(c, b)可以利用完全四點(diǎn)形的調和性質(zhì),解決問(wèn)題cos(a所以a,b調和分割c,d例2如圖1所示,AB為圓之直徑,C為直徑延長(cháng)線(xiàn)上一點(diǎn)從C向圓引切線(xiàn)CT,證明:若C在線(xiàn)段AB本身上,如何作它的調和共軛點(diǎn)?完全四點(diǎn)形:如圖3,平面內無(wú)三點(diǎn)共線(xiàn)的四點(diǎn)及其兩兩聯(lián)線(xiàn)所D構成的圖形稱(chēng)為完全四點(diǎn)形.并且完全四點(diǎn)形通過(guò)每一個(gè)對角點(diǎn)有組調和線(xiàn)束,即通過(guò)這對對角點(diǎn)的兩邊和對角三角形的兩條邊?，F在給出第四點(diǎn)D作法:如圖4.①過(guò)A,B,C作線(xiàn)外一點(diǎn)S,作線(xiàn)束SA,SB,SC作圖:如圖2,過(guò)點(diǎn)C作TT⊥AB,直接作TD⊥OT交AB延②在SC上取一點(diǎn)Q,并 AQXSB=T, BQX SA=R長(cháng)線(xiàn)于點(diǎn)D,即D調和分割定點(diǎn)A,B,C③作RT×ABC于點(diǎn)D,即D點(diǎn)為(AB,CD)=-1的第證明:需證T在A(yíng)B上的垂直射影,D是C對于A(yíng)B的調和四點(diǎn)共軛點(diǎn)需證:A.Bb=-(Ab·B參考文[1]朱德樣,朱維宗[2]李文銘,初等幾H中國煤化工樓CNMHG①資助課題:映西省特色專(zhuān)業(yè)建設項目(2011-59);安康學(xué)院大學(xué)生科技創(chuàng )新項目(2011 AKXYDXS09)Q戶(hù)淑茹(1989-),女,陜西省商洛市人,現為安康學(xué)院數學(xué)系09級學(xué)生。229