勾(gou)股(gu)定(ding)(ding)理(li),是(shi)一(yi)個基(ji)本的(de)(de)幾何定(ding)(ding)理(li),指直(zhi)(zhi)角(jiao)(jiao)三角(jiao)(jiao)形的(de)(de)兩條直(zhi)(zhi)角(jiao)(jiao)邊的(de)(de)平方和等于斜邊的(de)(de)平方。中(zhong)國古代稱直(zhi)(zhi)角(jiao)(jiao)三角(jiao)(jiao)形為(wei)勾(gou)股(gu)形,并且直(zhi)(zhi)角(jiao)(jiao)邊中(zhong)較小(xiao)者(zhe)為(wei)勾(gou),另一(yi)長直(zhi)(zhi)角(jiao)(jiao)邊為(wei)股(gu),斜邊為(wei)弦,所以稱這個定(ding)(ding)理(li)為(wei)勾(gou)股(gu)定(ding)(ding)理(li),也有(you)人稱商高定(ding)(ding)理(li)。
勾(gou)股(gu)定(ding)(ding)理現(xian)約有500種(zhong)證(zheng)明方法(fa),是數(shu)學定(ding)(ding)理中證(zheng)明方法(fa)最多的(de)(de)定(ding)(ding)理之(zhi)(zhi)(zhi)一。勾(gou)股(gu)定(ding)(ding)理是人類早期發現(xian)并證(zheng)明的(de)(de)重(zhong)要數(shu)學定(ding)(ding)理之(zhi)(zhi)(zhi)一,用代數(shu)思想解決幾何問題(ti)的(de)(de)最重(zhong)要的(de)(de)工具之(zhi)(zhi)(zhi)一,也是數(shu)形結合(he)的(de)(de)紐帶之(zhi)(zhi)(zhi)一。
在(zai)中國,周朝(chao)時期的(de)(de)商高提出了(le)“勾三股四弦五”的(de)(de)勾股定(ding)理的(de)(de)特例。在(zai)西方,最早(zao)提出并證(zheng)明此定(ding)理的(de)(de)為公元前6世紀古希臘的(de)(de)畢達哥拉斯學派(pai),他們用(yong)演繹法證(zheng)明了(le)直角三角形斜邊平方等于兩直角邊平方之和。
在平(ping)面上(shang)的一個直(zhi)角(jiao)三角(jiao)形中,兩個直(zhi)角(jiao)邊(bian)邊(bian)長的平(ping)方加(jia)起來等(deng)于斜(xie)邊(bian)長的平(ping)方。如果(guo)設直(zhi)角(jiao)三角(jiao)形的兩條(tiao)直(zhi)角(jiao)邊(bian)長度分(fen)別是(shi)和(he),斜(xie)邊(bian)長度是(shi),那么(me)可以用數學語(yu)言表達:
勾股定理是余弦定理中的一個特例。
《周髀算(suan)經(jing)》中,趙爽描述此圖(tu):“勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)各(ge)自乘(cheng)(cheng),并(bing)(bing)(bing)之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。開(kai)方(fang)(fang)(fang)(fang)除(chu)之(zhi)(zhi)(zhi)(zhi)(zhi),即(ji)(ji)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)。案玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)圖(tu)有可(ke)以(yi)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)相(xiang)(xiang)乘(cheng)(cheng)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)朱實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)二,倍(bei)(bei)之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)朱實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)四。以(yi)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)之(zhi)(zhi)(zhi)(zhi)(zhi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)自相(xiang)(xiang)乘(cheng)(cheng)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)中黃(huang)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。加(jia)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)亦(yi)(yi)成玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)減玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi),半其(qi)(qi)(qi)余(yu)(yu)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)從法(fa)(fa)(fa),開(kai)方(fang)(fang)(fang)(fang)除(chu)之(zhi)(zhi)(zhi)(zhi)(zhi),復得(de)(de)(de)(de)勾(gou)(gou)(gou)(gou)(gou)(gou)矣。加(jia)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)于勾(gou)(gou)(gou)(gou)(gou)(gou)即(ji)(ji)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)。凡并(bing)(bing)(bing)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)之(zhi)(zhi)(zhi)(zhi)(zhi)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi),即(ji)(ji)成玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。或(huo)(huo)矩(ju)(ju)(ju)于內,或(huo)(huo)方(fang)(fang)(fang)(fang)于外。形詭而量均,體殊而數齊。勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)之(zhi)(zhi)(zhi)(zhi)(zhi)矩(ju)(ju)(ju)以(yi)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)廣(guang)(guang),股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)并(bing)(bing)(bing)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)袤(mao)。而股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)方(fang)(fang)(fang)(fang)其(qi)(qi)(qi)里(li)(li)。減矩(ju)(ju)(ju)勾(gou)(gou)(gou)(gou)(gou)(gou)之(zhi)(zhi)(zhi)(zhi)(zhi)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)于玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi),開(kai)其(qi)(qi)(qi)余(yu)(yu)即(ji)(ji)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)。倍(bei)(bei)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)在(zai)兩(liang)(liang)邊為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)從法(fa)(fa)(fa),開(kai)矩(ju)(ju)(ju)勾(gou)(gou)(gou)(gou)(gou)(gou)之(zhi)(zhi)(zhi)(zhi)(zhi)角(jiao)(jiao)即(ji)(ji)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。加(jia)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)除(chu)勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)得(de)(de)(de)(de)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)并(bing)(bing)(bing)。以(yi)并(bing)(bing)(bing)除(chu)勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)亦(yi)(yi)得(de)(de)(de)(de)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。令并(bing)(bing)(bing)自乘(cheng)(cheng)與(yu)勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。倍(bei)(bei)并(bing)(bing)(bing)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)法(fa)(fa)(fa)。所(suo)得(de)(de)(de)(de)亦(yi)(yi)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)。勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)減并(bing)(bing)(bing)自乘(cheng)(cheng),如法(fa)(fa)(fa)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)。股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)之(zhi)(zhi)(zhi)(zhi)(zhi)矩(ju)(ju)(ju)以(yi)勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)廣(guang)(guang),勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)并(bing)(bing)(bing)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)袤(mao)。而勾(gou)(gou)(gou)(gou)(gou)(gou)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)方(fang)(fang)(fang)(fang)其(qi)(qi)(qi)里(li)(li),減矩(ju)(ju)(ju)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)之(zhi)(zhi)(zhi)(zhi)(zhi)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)于玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi),開(kai)其(qi)(qi)(qi)余(yu)(yu)即(ji)(ji)勾(gou)(gou)(gou)(gou)(gou)(gou)。倍(bei)(bei)勾(gou)(gou)(gou)(gou)(gou)(gou)在(zai)兩(liang)(liang)邊為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)從法(fa)(fa)(fa),開(kai)矩(ju)(ju)(ju)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)之(zhi)(zhi)(zhi)(zhi)(zhi)角(jiao)(jiao),即(ji)(ji)勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。加(jia)勾(gou)(gou)(gou)(gou)(gou)(gou)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)除(chu)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)得(de)(de)(de)(de)勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)并(bing)(bing)(bing)。以(yi)并(bing)(bing)(bing)除(chu)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)亦(yi)(yi)得(de)(de)(de)(de)勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。令并(bing)(bing)(bing)自乘(cheng)(cheng)與(yu)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。倍(bei)(bei)并(bing)(bing)(bing)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)法(fa)(fa)(fa)。所(suo)得(de)(de)(de)(de)亦(yi)(yi)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)。股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)減并(bing)(bing)(bing)自乘(cheng)(cheng)如法(fa)(fa)(fa)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)勾(gou)(gou)(gou)(gou)(gou)(gou),兩(liang)(liang)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)相(xiang)(xiang)乘(cheng)(cheng)倍(bei)(bei)而開(kai)之(zhi)(zhi)(zhi)(zhi)(zhi),所(suo)得(de)(de)(de)(de)以(yi)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)增之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)勾(gou)(gou)(gou)(gou)(gou)(gou)。以(yi)勾(gou)(gou)(gou)(gou)(gou)(gou)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)增之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)。兩(liang)(liang)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)增之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)弦(xian)。倍(bei)(bei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)列勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi),見(jian)并(bing)(bing)(bing)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)者,以(yi)圖(tu)考之(zhi)(zhi)(zhi)(zhi)(zhi),倍(bei)(bei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)滿外大方(fang)(fang)(fang)(fang)而多(duo)(duo)黃(huang)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。黃(huang)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)之(zhi)(zhi)(zhi)(zhi)(zhi)多(duo)(duo),即(ji)(ji)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)減之(zhi)(zhi)(zhi)(zhi)(zhi),開(kai)其(qi)(qi)(qi)余(yu)(yu),得(de)(de)(de)(de)外大方(fang)(fang)(fang)(fang)。大方(fang)(fang)(fang)(fang)之(zhi)(zhi)(zhi)(zhi)(zhi)面(mian)(mian),即(ji)(ji)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)并(bing)(bing)(bing)也。令并(bing)(bing)(bing)自乘(cheng)(cheng),倍(bei)(bei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)乃減之(zhi)(zhi)(zhi)(zhi)(zhi),開(kai)其(qi)(qi)(qi)余(yu)(yu),得(de)(de)(de)(de)中黃(huang)方(fang)(fang)(fang)(fang)。黃(huang)方(fang)(fang)(fang)(fang)之(zhi)(zhi)(zhi)(zhi)(zhi)面(mian)(mian),即(ji)(ji)勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)減并(bing)(bing)(bing)而半之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)勾(gou)(gou)(gou)(gou)(gou)(gou)。加(jia)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)于并(bing)(bing)(bing)而半之(zhi)(zhi)(zhi)(zhi)(zhi)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)。其(qi)(qi)(qi)倍(bei)(bei)玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)廣(guang)(guang)袤(mao)合。令勾(gou)(gou)(gou)(gou)(gou)(gou)股(gu)(gu)(gu)(gu)(gu)(gu)(gu)(gu)見(jian)者自乘(cheng)(cheng)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)其(qi)(qi)(qi)實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)。四實(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)(shi)以(yi)減之(zhi)(zhi)(zhi)(zhi)(zhi),開(kai)其(qi)(qi)(qi)余(yu)(yu),所(suo)得(de)(de)(de)(de)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)。以(yi)差(cha)(cha)(cha)(cha)(cha)(cha)(cha)(cha)減合半其(qi)(qi)(qi)余(yu)(yu)為(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)(wei)廣(guang)(guang)。減廣(guang)(guang)于玄(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)(xuan)即(ji)(ji)所(suo)求(qiu)也。”
用(yong)現代的數學語(yu)言描述就是黃(huang)實的面積等于大正(zheng)方形的面積減(jian)去四(si)個(ge)朱(zhu)實的面積。
2002年第24屆國際(ji)數學家大會(hui)(hui)(ICM)的會(hui)(hui)標即為該圖(tu)。
加菲爾德在(zai)證出(chu)此結(jie)論5年后,成為美國第20任(ren)總統,所以人們又稱(cheng)其為“總統證法”。
在直角梯形(xing)ABDE中,∠AEC=∠CDB=90°,△AEC≌△CDB,,,
∵
加(jia)菲(fei)爾德證法變式
該證(zheng)明(ming)為加菲爾德證(zheng)法的變式。
如(ru)果將(jiang)大正(zheng)方形邊長為(wei)c的小正(zheng)方形沿對(dui)角線切開,則回到了(le)加菲爾德證(zheng)法。相反,若(ruo)將(jiang)上圖(tu)中兩個梯形拼在一起(qi),就變為(wei)了(le)此(ci)證(zheng)明方法。
大正(zheng)方(fang)形的(de)(de)面(mian)(mian)積等于中間正(zheng)方(fang)形的(de)(de)面(mian)(mian)積加上(shang)四個三角形的(de)(de)面(mian)(mian)積,即:
青朱出入圖,是東(dong)漢末年(nian)數學家劉徽根據(ju)“割補術”運用數形關系(xi)證明勾(gou)股定理的幾何證明法,特色鮮明、通俗易懂。
劉徽描述此圖(tu),“勾自乘為朱方(fang)(fang),股(gu)自乘為青方(fang)(fang),令出入相補(bu),各從其類(lei),因就(jiu)其余不動也,合成弦(xian)方(fang)(fang)之(zhi)冪(mi)。開(kai)方(fang)(fang)除(chu)之(zhi),即弦(xian)也。”其大意(yi)為,一個(ge)任(ren)意(yi)直角三角形(xing)(xing),以(yi)勾寬作紅色正(zheng)(zheng)方(fang)(fang)形(xing)(xing)即朱方(fang)(fang),以(yi)股(gu)長(chang)作青色正(zheng)(zheng)方(fang)(fang)形(xing)(xing)即青方(fang)(fang)。將朱方(fang)(fang)、青方(fang)(fang)兩個(ge)正(zheng)(zheng)方(fang)(fang)形(xing)(xing)對(dui)齊底邊排列,再以(yi)盈補(bu)虛,分(fen)割線內(nei)不動,線外則“各從其類(lei)”,以(yi)合成弦(xian)的正(zheng)(zheng)方(fang)(fang)形(xing)(xing)即弦(xian)方(fang)(fang),弦(xian)方(fang)(fang)開(kai)方(fang)(fang)即為弦(xian)長(chang)。
在歐幾(ji)里得的《幾(ji)何原本》一(yi)(yi)書中給出勾股定理的以(yi)下證明。設△ABC為一(yi)(yi)直(zhi)角(jiao)三角(jiao)形,其(qi)(qi)中A為直(zhi)角(jiao)。從A點(dian)畫一(yi)(yi)直(zhi)線(xian)至(zhi)對邊(bian),使其(qi)(qi)垂直(zhi)于對邊(bian)。延長此線(xian)把對邊(bian)上(shang)的正方形一(yi)(yi)分為二,其(qi)(qi)面(mian)積分別與其(qi)(qi)余兩個正方形相等。
在(zai)這個定理的證明(ming)中(zhong),我們需要如下四個輔(fu)助定理:
如果兩個三(san)角形有(you)兩組(zu)對應邊和這(zhe)兩組(zu)邊所夾的角相等,則兩三(san)角形全等。(SAS)
三角(jiao)形(xing)面(mian)(mian)積(ji)是任一同底同高之平行四邊(bian)形(xing)面(mian)(mian)積(ji)的一半。
任意一個正方形(xing)的面積等于其二邊(bian)長的乘積。
任意一(yi)個(ge)矩形的面(mian)積(ji)等于其二(er)邊(bian)長的乘積(ji)(據(ju)輔助定理3)。
證明的(de)思(si)路為:從A點畫一直線(xian)至對邊,使其垂直于(yu)對邊。延長此線(xian)把對邊上的(de)正方(fang)形(xing)一分為二,把上方(fang)的(de)兩(liang)個正方(fang)形(xing),通(tong)過等高同(tong)底的(de)三角(jiao)形(xing),以其面(mian)積關系,轉(zhuan)換成下方(fang)兩(liang)個同(tong)等面(mian)積的(de)長方(fang)形(xing)。
設△ABC為一(yi)直(zhi)角三角形,其直(zhi)角為∠CAB。
其邊為(wei)BC、AB和CA,依(yi)序繪成(cheng)四方形CBDE、BAGF和ACIH。
畫(hua)出(chu)過點A之(zhi)BD、CE的(de)平行線,分別(bie)垂直BC和(he)DE于K、L。
分別連接CF、AD,形成△BCF、△BDA。
∠CAB和(he)∠BAG都是(shi)直角,因此(ci)C、A和(he)G共線,同理(li)可證(zheng)B、A和(he)H共線。
∠CBD和∠FBA都是直角,所以∠ABD=∠FBC。
因(yin)為AB=FB,BD=BC,所以△ABD≌△FBC。
因為A與K和(he)L在同一直線上(shang),所以四邊形BDLK=2△ABD。
因為C、A和G在同一直線(xian)上,所以正方(fang)形BAGF=2△FBC。
因(yin)此四(si)邊形BDLK=BAGF=AB2。
同理可(ke)證,四邊形CKLE=ACIH=AC2。
把這兩個結(jie)果相加,AB2+AC2=BD×BK+KL×KC
由于BD=KL,BD×BK+KL×KC=BD(BK+KC)=BD×BC
由于CBDE是個正方(fang)形,因(yin)此(ci)AB2+AC2=BC2,即a2+b2=c2。
此證(zheng)明是于歐幾里得《幾何原本》一(yi)書第1.47節所提出(chu)的。
由于(yu)這(zhe)個定(ding)理(li)(li)的(de)證明依賴于(yu)平(ping)行(xing)公(gong)理(li)(li),而且從這(zhe)個定(ding)理(li)(li)可以推出平(ping)行(xing)公(gong)理(li)(li),很多(duo)人質疑平(ping)行(xing)公(gong)理(li)(li)是這(zhe)個定(ding)理(li)(li)的(de)必要條件,一(yi)直到(dao)十九世(shi)紀嘗試否定(ding)第五公(gong)理(li)(li)的(de)非歐(ou)幾何出現。
勾(gou)(gou)股數(shu)(shu)組(zu)是滿足(zu)勾(gou)(gou)股定理(li)的正整數(shu)(shu)組(zu),其中(zhong)的稱為勾(gou)(gou)股數(shu)(shu)。例如就(jiu)是一組(zu)勾(gou)(gou)股數(shu)(shu)組(zu)。
任(ren)意一組勾股數可以表示為如(ru)下形(xing)式:,,,其中均(jun)為正(zheng)整數,且。
已知直(zhi)角(jiao)三(san)角(jiao)形(xing)(xing)兩邊(bian)(bian)求(qiu)解(jie)第三(san)邊(bian)(bian),或者已知三(san)角(jiao)形(xing)(xing)的三(san)邊(bian)(bian)長(chang)(chang)度,證明該(gai)三(san)角(jiao)形(xing)(xing)為直(zhi)角(jiao)三(san)角(jiao)形(xing)(xing)或用來證明該(gai)三(san)角(jiao)形(xing)(xing)內兩邊(bian)(bian)垂直(zhi)。利用勾股(gu)定理求(qiu)線段(duan)長(chang)(chang)度這是勾股(gu)定理的最基(ji)本運用。
公元前(qian)十一世紀,數學家商(shang)高(gao)(西周初年人(ren))就(jiu)提出“勾(gou)(gou)三(san)、股四(si)、弦五(wu)”。編(bian)寫(xie)于(yu)公元前(qian)一世紀以前(qian)的(de)《周髀算經(jing)》中記(ji)錄著(zhu)商(shang)高(gao)與周公的(de)一段(duan)對話。商(shang)高(gao)說(shuo):“……故折矩,勾(gou)(gou)廣三(san),股修(xiu)四(si),經(jing)隅五(wu)。”意為(wei)(wei):當(dang)直角(jiao)三(san)角(jiao)形的(de)兩條直角(jiao)邊分別為(wei)(wei)3(勾(gou)(gou))和(he)4(股)時,徑隅(弦)則為(wei)(wei)5。以后人(ren)們(men)就(jiu)簡單地把(ba)這(zhe)個事(shi)實(shi)說(shuo)成“勾(gou)(gou)三(san)股四(si)弦五(wu)”,根據(ju)該典故稱勾(gou)(gou)股定(ding)理為(wei)(wei)商(shang)高(gao)定(ding)理。
公元三世紀,三國時代的(de)趙(zhao)爽對《周(zhou)髀算經(jing)》內的(de)勾(gou)股(gu)定(ding)理作出了(le)詳(xiang)(xiang)細注釋,記錄于(yu)《九章(zhang)算術》中“勾(gou)股(gu)各自乘(cheng),并而開方除之,即弦”,趙(zhao)爽創(chuang)制了(le)一幅“勾(gou)股(gu)圓方圖”,用數形結合得(de)到方法,給出了(le)勾(gou)股(gu)定(ding)理的(de)詳(xiang)(xiang)細證(zheng)(zheng)明。后劉徽(hui)(hui)在劉徽(hui)(hui)注中亦證(zheng)(zheng)明了(le)勾(gou)股(gu)定(ding)理。
在(zai)中國清朝末(mo)年,數學(xue)家華蘅芳提(ti)出(chu)了二十(shi)多種對于勾股定(ding)理證法。
遠在(zai)公(gong)元前約三千年的古(gu)巴比(bi)倫(lun)人就知道(dao)和應用勾股定理,他們還知道(dao)許(xu)多勾股數組。美國哥倫(lun)比(bi)亞大學圖(tu)書館(guan)內(nei)收(shou)藏著(zhu)一塊(kuai)編(bian)號為“普(pu)林頓322”的古(gu)巴比(bi)倫(lun)泥板,上面就記載(zai)了很多勾股數。古(gu)埃及(ji)人在(zai)建(jian)筑(zhu)宏偉(wei)的金(jin)字(zi)塔和測(ce)量(liang)尼羅河泛濫后的土地時,也應用過(guo)勾股定理。
公元前六世紀,希臘數學家畢達(da)哥(ge)拉斯(si)證明了勾(gou)股(gu)定理,因(yin)而西方人都習(xi)慣(guan)地稱這個定理為畢達(da)哥(ge)拉斯(si)定理。
公元前4世紀,希臘數(shu)學家歐幾(ji)里(li)得在《幾(ji)何原本(ben)》(第Ⅰ卷,命題47)中給(gei)出一(yi)個(ge)證明(ming)。
1876年4月1日,加菲(fei)爾德在《新英格蘭(lan)教(jiao)育日志》上發表了他對勾股(gu)定(ding)理的(de)一個證法(fa)。
1940年《畢達哥拉(la)斯命題》出版,收集了367種不(bu)同的證法。
1.勾股(gu)定理的證明是論證幾何的發端。
2.勾(gou)股(gu)定(ding)理(li)是(shi)歷史上第一個把(ba)數與(yu)形(xing)聯(lian)系起(qi)來的定(ding)理(li),即它是(shi)第一個把(ba)幾(ji)何與(yu)代(dai)數聯(lian)系起(qi)來的定(ding)理(li)。
3.勾股定(ding)理導致了無理數的(de)發現,引起第一次數學危機,大大加深(shen)了人們(men)對數的(de)理解。
4.勾股(gu)定(ding)理是歷(li)史上第一個給(gei)出了完全解答(da)的不定(ding)方(fang)程,它(ta)引出了費馬大定(ding)理。
5.勾(gou)股(gu)定(ding)理是歐氏幾何(he)的(de)(de)基(ji)礎定(ding)理,并有(you)巨大的(de)(de)實用(yong)價值。這(zhe)條(tiao)定(ding)理不(bu)僅在(zai)幾何(he)學(xue)(xue)中是一(yi)顆光彩奪(duo)目的(de)(de)明(ming)珠(zhu),被譽為(wei)“幾何(he)學(xue)(xue)的(de)(de)基(ji)石”,而且在(zai)高等數(shu)(shu)(shu)學(xue)(xue)和其他科學(xue)(xue)領域也有(you)著廣泛的(de)(de)應用(yong)。1971年(nian)5月(yue)15日,尼加拉瓜發行了一(yi)套題為(wei)“改變世界面貌的(de)(de)十個(ge)數(shu)(shu)(shu)學(xue)(xue)公式”郵票,這(zhe)十個(ge)數(shu)(shu)(shu)學(xue)(xue)公式由著名數(shu)(shu)(shu)學(xue)(xue)家選出的(de)(de),勾(gou)股(gu)定(ding)理是其中之首。