2012年2月觀周刊第8期總568期.February 2012 M i有關(guān)調(diào)和性質(zhì)的應(yīng)用°戶淑茹②趙臨龍(安康學(xué)院數(shù)學(xué)與應(yīng)用數(shù)學(xué)研究所安康725000中圖分類號:0185.1文獻(xiàn)標(biāo)識碼:A文章編號:1008-925X(2012)02-0229-01摘要:利用調(diào)和分割的定義解決初等幾何問題:如何做調(diào)和共軛點(diǎn)、證明一角的兩邊和它的內(nèi)外交平分線成調(diào)和線束、一線段被它的中點(diǎn)和這直線上的無穿遠(yuǎn)點(diǎn)所調(diào)和分割、以及如何用完全四點(diǎn)形的調(diào)和性質(zhì)作圖關(guān)鍵詞:調(diào)和分割中點(diǎn)無窮遠(yuǎn)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(diào)和分割的定義:設(shè)A,B,C,D為共線的四點(diǎn),我們把由于∠2=∠TOB,∠1=∠BOTAG:B犯定義為這四點(diǎn)按著順序的交比,并以符號(ABCD)∴∠1=∠2,同理:∠3=∠4表達(dá)CAB, CD)=B當(dāng)(AB,CD)=-1時,即C點(diǎn)分割線(由三角形中一個角的內(nèi)角和外交平分線和對邊的交點(diǎn),調(diào)和分割對點(diǎn)段AB的值和D點(diǎn)分割線段AB的值只差一個符號(因而一個是從中學(xué)幾何,我們知道三角形中一個角的內(nèi)角和外角平分線內(nèi)分點(diǎn),一個是外分點(diǎn)),這時說C,D兩點(diǎn)調(diào)和分割線段AB或?qū)叿殖蓛删€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于線段AB成調(diào)和共軛點(diǎn)偶是絕對量!),可見兩條平分角線和對邊的交點(diǎn),調(diào)和分割對邊但是,在調(diào)和分割中,兩對點(diǎn)的關(guān)系是完全對等的,即當(dāng)C與例3一線段被它的中點(diǎn)和這直線上的無窮遠(yuǎn)點(diǎn)所調(diào)和D調(diào)和分割線段AB時,A與B也調(diào)和分割CD.若(AB,CD)=分割。1,便也有(CD,AB)=-1。證明:設(shè)C點(diǎn)為線段AB的中點(diǎn),D。為直線AB的無窮遠(yuǎn)點(diǎn)例1試證一角的兩邊和它的內(nèi)外交平分線成調(diào)和線束。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,共線四點(diǎn)的交比,若第四點(diǎn)為無窮遠(yuǎn)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)的簡比值sin(a,d)·sin(b,c)sin[90°+(a,c)]例4調(diào)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(diào)和性質(zhì),解決問題cos(a所以a,b調(diào)和分割c,d例2如圖1所示,AB為圓之直徑,C為直徑延長線上一點(diǎn)從C向圓引切線CT,證明:若C在線段AB本身上,如何作它的調(diào)和共軛點(diǎn)?完全四點(diǎn)形:如圖3,平面內(nèi)無三點(diǎn)共線的四點(diǎn)及其兩兩聯(lián)線所D構(gòu)成的圖形稱為完全四點(diǎn)形.并且完全四點(diǎn)形通過每一個對角點(diǎn)有組調(diào)和線束,即通過這對對角點(diǎn)的兩邊和對角三角形的兩條邊?，F(xiàn)在給出第四點(diǎn)D作法:如圖4.①過A,B,C作線外一點(diǎn)S,作線束SA,SB,SC作圖:如圖2,過點(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長線于點(diǎn)D,即D調(diào)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AB上的垂直射影,D是C對于AB的調(diào)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:映西省特色專業(yè)建設(shè)項(xiàng)目(2011-59);安康學(xué)院大學(xué)生科技創(chuàng)新項(xiàng)目(2011 AKXYDXS09)Q戶淑茹(1989-),女,陜西省商洛市人,現(xiàn)為安康學(xué)院數(shù)學(xué)系09級學(xué)生。229