已知A、B、C是半径为1的球面上三点，O为球心，A、B和A、C的球面距离都是π/2，B、C的球面距离是π/3 已知球O的半径为1，A、B、C三点都在球面上，且每两点间的球...
\u534a\u5f84\u4e3a1\u7684\u7403\u9762\u4e0a\u6709A\uff0cB\uff0cC\u4e09\u70b9\uff0c\u5176\u4e2dA\u548cB\u7684\u7403\u9762\u8ddd\u79bb\uff0cA\u548cC\u7684\u7403\u9762\u8ddd\u79bb\u90fd\u662f\u03c02\uff0cB\u548cC\u7684\u7403\u9762\u8ddd\u79bb\u662f\u03c03\uff0e\uff081\uff081\uff09\u7531\u9898\u610f\u77e5\uff1a\u2220AOC=\u03c02\uff0c\u2220AOB=\u03c02\uff0c\u2220BOC=\u03c03\uff0c\u2234AO\u22a5\u9762BOC\u2235OA=OB=OC=1\uff0c\u2234AB=AC=2\uff0cBC=1\uff0e\u2235VA?OBC\uff1d13S\u25b3OBC|AO|\uff1d312\u53c8VA?OBC\uff1d13S\u25b3ABC?h\uff08h\u4e3aO\u5230\u5e73\u9762ABC\u7684\u8ddd\u79bb\uff09\u2235S\u25b3ABC\uff1d74\u2234h\uff1d217\u2234\u7403\u5fc3O\u5230\u5e73\u9762ABC\u7684\u8ddd\u79bb217\uff084\u5206\uff09\uff082\uff09\u8fc7O\u4f5cOD\u22a5BC\uff0c\u2235AO\u22a5\u9762BOC\uff0c\u4e14OD?\u9762BOC\uff0c\u2234OD\u22a5AO\uff0c\u2234OD\u4e3a\u5f02\u9762\u76f4\u7ebfOA\u548cBC\u7684\u516c\u5782\u7ebf\u6bb5\uff0c\u5373\u4e3a\u5f02\u9762\u76f4\u7ebfOA\u548cBC\u7684\u8ddd\u79bb\uff0e\u53c8\u2235\u25b3OBC\u4e3a\u7b49\u8fb9\u4e09\u89d2\u5f62\uff0c\u4e14\u8fb9\u957f\u4e3a1\uff0e\u6240\u4ee5OD=<div style="width: 6px; background-image: url(http://hiphotos.baidu.com/zhidao/pic/item/aa64034f78f0f736dcbbf8b50955b319ebc41338.jpg); background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; overflow-x: hidden; overflow-y: hidden; height: 11px; background-position: initial initi
\u663e\u7136OA\u3001OB\u3001OC\u4e24\u4e24\u5782\u76f4\uff0c\u5982\u56fe\uff0c\u8bbeO 1 \u4e3aABC\u6240\u5728\u5e73\u9762\u622a\u7403\u6240\u5f97\u5706\u7684\u5706\u5fc3\uff0c\u2235OA=OB=OC=1\uff0c\u4e14OA\u22a5OB\u22a5OC\uff0c\u2234AB=BC=CA= 2 \uff0e\u2234O 1 \u4e3a\u25b3ABC\u7684\u4e2d\u5fc3\uff0e\u2234O 1 A= 6 3 \uff0e\u7531OO 1 2 +O 1 A 2 =OA 2 \uff0c\u53ef\u5f97OO 1 = 3 3 \uff0e\u6545\u9009B\uff0e
球半径R=1角AOB=π/2,AB=2sin(π/4)=根号2,AC=AB=根号2
角BOC=π/3,BC=2sin(π/6)=1
sin(角BAC/2)=(1/2)/(根号2)=1/(2(根号2))=(根号2)/4
cos(角BAC/2)=(1-((根号2)/4)^2)^(1/2)=(1/4)(根号(14)
sin(角BAC)=2sin(角BAC/2)cos(角BAC/2)=(1/4)(根号7)
设三角形ABC的外接圆半径=r
2r=BC/sin(角BAC)=4/(根号7)
r=2/(根号7)
球心O到平面ABC的距离=(R^2-r^2)^(1/2)=(1-(4/7))^(1/2)=(1/7)(根号21)
AB=AC=√2
BC=1
等腰△ABC的外心到A的距离=2√7/7
心O到平面ABC的距离²=1²-（2√7/7）²=21/49
心O到平面ABC的距离=√21/7
绛旓細鐞冨崐寰R=1 瑙扐OB=蟺/2,AB=2sin(蟺/4)=鏍瑰彿2,AC=AB=鏍瑰彿2 瑙払OC=蟺/3,BC=2sin(蟺/6)=1 sin(瑙払AC/2)=(1/2)/(鏍瑰彿2)=1/(2(鏍瑰彿2))=(鏍瑰彿2)/4 cos(瑙払AC/2)=(1-((鏍瑰彿2)/4)^2)^(1/2)=(1/4)(鏍瑰彿(14)sin(瑙払AC)=2sin(瑙払AC/2)cos(瑙払AC/2)=(1/4)(...
绛旓細瑙ｏ細濡傚浘锛岋紙1锛夊洜涓虹悆O鐨鍗婂緞涓1锛孊銆丆涓ょ偣闂鐨勭悆闈璺濈涓合3锛岀偣A涓嶣銆丆涓ょ偣闂寸殑鐞冮潰璺濈鍧囦负蟺2锛屾墍浠モ垹BOC=蟺3锛屸垹AOB=鈭燗OC=蟺2锛庯紙2锛夊洜涓BC=1锛孉C=AB=2锛屾墍浠ョ敱浣欏鸡瀹氱悊寰梒os鈭燘AC=34锛宻in鈭燘AC=74锛岃鎴潰鍦嗙殑鍦嗗績涓篛1锛岃繛鎺O1锛屽垯鎴潰鍦嗙殑鍗婂緞R=AO1锛岀敱姝ｅ鸡瀹氱悊寰梤=BC2s...
绛旓細閫氳繃宸茬煡A銆丅銆C鏄崐寰勪负1鐨勭悆闈涓婄殑涓夌偣鍙煡OA=OB=OC=1銆侭銆丆涓ょ偣闂寸殑鐞冮潰璺濈涓合/3锛屽彲鐭モ垹BOC=60掳锛孊C=1 鐐笰涓嶣銆丆涓ょ偣闂寸殑鐞冮潰璺濈鍧囦负蟺/2锛屽彲鐭モ垹AOB=鈭燗OC=90掳 浜﹀彲鎺ㄥ嚭OA鈯ラ潰OBC銆侽鍒版埅闈BC鐨勮窛绂伙紝鍗宠繃O鍚戔柍ABC鍋氱殑鍨傜嚎璺濈锛岄氳繃鈻抽潰绉彲浠ョ畻鍑篛鍒版埅闈BC鐨勮窛绂讳负锛...
绛旓細鍏堟眰鍑哄鸡闀夸负1锛60搴﹀渾蹇冭鍗婂緞绛変簬寮﹂暱锛孉B=AC=BC=1锛屽湪绛夎竟涓夎褰BC涓叾楂樹负鈭3/2锛屽唴蹇冧笌椤剁偣璺濈涓衡垰3/3锛岀悆蹇僌鍒板钩闈BC鐨勮窛绂=鈭1^2-(鈭3/3)^2=鈭6/3.
绛旓細璁綩(0,0,0),A(0,0,1),B(1,0,0),C(1/2,sqrt(3)/2,0)V(O-ABC)=1/3*1*sqrt(3)/4=sqrt(3)/12 S(ABC)=1/2|ABxAC| =1/2|(sqrt(3)/2,,1/2,sqrt(3)/2)| =7/8 d=(sqrt(3)/4)/(7/8)=2sqrt(3)/7 ...
绛旓細OBC锛13S鈻砄BC|AO|锛312鍙圴A?OBC锛13S鈻ABC?h锛坔涓篛鍒板钩闈BC鐨璺濈锛夆埖S鈻矨BC锛74鈭磆锛217鈭寸悆蹇僌鍒板钩闈BC鐨勮窛绂217锛4鍒嗭級锛2锛夎繃O浣淥D鈯BC锛鈭礎O鈯闈OC锛屼笖OD?闈OC锛屸埓OD鈯O锛屸埓OD涓哄紓闈㈢洿绾縊A鍜孊C鐨勫叕鍨傜嚎娈碉紝鍗充负寮傞潰鐩寸嚎OA鍜孊C鐨勮窛绂伙紟鍙堚埖鈻砄BC涓虹瓑杈逛笁瑙掑舰锛屼笖...
绛旓細鐞冨績O涓A锛孊锛孋涓夌偣鏋勬垚姝ｄ笁妫遍敟O-ABC锛宸茬煡OA=OB=OC=R=1锛屸垹AOB=鈭燘OC=鈭燗OC=90掳锛岀敱姝ゅ彲寰桝O鈯闈OC锛庘埖 S 鈻矪OC = 1 2 锛 S 鈻矨BC = 3 2 锛庤鐞冨績鍒伴潰鐨勮窛绂讳负h锛岀敱V A-BOC =V O-ABC 锛屽緱 h= 3 3 锛庢墍浠ョ悆蹇僌 鍒板钩闈...
绛旓細瑙ｏ細鐞冨績O涓A锛孊锛孋涓夌偣鏋勬垚姝ｄ笁妫遍敟O-ABC锛屽鍥炬墍绀猴紝宸茬煡OA=OB=OC=R=1锛屸垹AOB=鈭燘OC=鈭燗OC=90掳锛岀敱姝ゅ彲寰桝O鈯闈OC锛庘埖S鈻矪OC锛12锛孲鈻矨BC锛32锛庘埓鐢盫A-BOC=VO-ABC锛屽緱h锛33锛庢晠绛旀涓猴細33
绛旓細瑙ｏ細鈭礎銆丆涓ょ偣鐨勭悆闈璺濈涓合2锛屸埓AC=鈭2 鈭礎B=1锛孊C=鈭3锛孉C=鈭2锛屸埓鈻矨BC涓浠A涓鐩磋鐨勭洿瑙掍笁瑙掑舰 鈭村钩闈BC鎴悆寰楀埌鐨勬埅闈㈠渾鍗婂緞r=12BC=鈭32 鈭寸悆蹇僌鍒板钩闈BC鐨勮窛绂籨=鈭歊2-r2=12 鏁呴塁锛
绛旓細1:褰撲笁涓笁瑙掑舰閮芥槸绛夎叞鐩磋涓夎褰㈡椂,闈㈢Н涔嬪拰鏈澶 2:涓涓崐寰勪负2鐨勭悆鐨勫唴鎺ユ鏂逛綋鐨勭浉閭3涓潰涓や袱浜掔浉鍨傜洿,鐩搁偦3涓竟灏辨槸ab,ac,ad 3:鐞冪殑鍗婂緞涓鍐呮帴姝ｆ柟浣撶殑涓績鍒伴《鐐圭殑闀垮害 璁炬鏂逛綋杈归暱涓a,鍒 涓績鍒伴《鐐圭殑闀垮害=鈭(a^2+a^+a^2)/2=鐞冪殑鍗婂緞=1 鈭(a^2+a^2+a^2)=2 3a^2...
