求图：正四面体内切球与外接球的体积的比是多少？
\u6b63\u56db\u9762\u4f53\u7684\u5185\u5207\u7403\u548c\u5916\u63a5\u7403\u7684\u4f53\u79ef\u6bd4\uff01\u8fd9\u9053\u9898\u7684\u56fe\u600e\u6837\u753b\uff01\uff01\u4f53\u79ef\u6bd41:27
\u65b9\u6cd5\u4e00\uff1a
\u8bbe\u6b63\u56db\u9762\u4f53\u4e3aABCD\uff0c\u8fc7A\u505a\u5e95\u9762BCD\u7684\u5782\u7ebf\uff0c\u5782\u8db3\u4e3aM\uff0cM\u662f\u25b3BCD\u7684\u91cd\u5fc3\uff08\u4e09\u5fc3\u5408\u4e00\uff09\uff0c\u82e5\u8bbe\u8fb9\u957f\u4e3a1\uff0c\u5219\u53ef\u6c42\u5f97BM=2/3 * \u221a3/2 =\u221a3/3\uff0c\u5219AM=\u221a6/3,\u3002\u7136\u540e\u5728\u9762ABM\u4e2d\u505aAB\u7684\u5782\u76f4\u5e73\u5206\u7ebf\u4ea4AM\u4e8e\u70b9O\uff0c\u4ea4AB\u4e0e\u70b9N\u3002\u53ef\u8bc1\u660eAO=BO=CO=DO\uff0c\u4e14O\u5230\u9762ABC\uff0cBCD\uff0cCDA\u3001BCA\u7684\u8ddd\u79bb\u76f8\u7b49\u3002\u6240\u4ee5O\u5c31\u662f\u5916\u63a5\u7403\u8ddf\u5185\u5207\u7403\u7684\u7684\u7403\u5fc3\uff0c\u6545\u800c\u4f53\u79ef\u6bd4\u5c31\u662f\u534a\u5f84\u6bd4\u7684\u7acb\u65b9\u3002\u663e\u7136Rt\u25b3ABM\u223dRt\u25b3AON\uff0c\u6709AB/AM=AO/AN,2AN=AB.\u53ef\u6c42\u5f97AO=\u221a6/4\uff0c\u6545\u800cOM=AM-AO=\u221a6/3-\u221a6/4=\u221a6/12. \u6240\u4ee5\u534a\u5f84\u6bd4\u4e3a OM\uff1aAO= 1:3\uff0c\u6240\u4ee5\u4f53\u79ef\u6bd4\u4e3a1:27
\u65b9\u6cd5\u4e8c\uff1a
\u8bbe\u6b63\u56db\u9762\u4f53\u7684\u8fb9\u957f\u4e3a\u221a2a\uff0c\u5219\u8fd9\u4e2a\u6b63\u56db\u9762\u4f53\u53ef\u4ee5\u770b\u6210\u662f\u7531\u8fb9\u957f\u4e3aa\u7684\u6b63\u65b9\u4f53\u5207\u5272\u51fa\u6765\u7684\u3002
1\u3001\u6b63\u56db\u9762\u4f53\u7684\u5916\u63a5\u7403\u534a\u5f84R\u5c31\u662f\u6b63\u65b9\u4f53\u5bf9\u89d2\u7ebf\u76842\u5206\u4e4b1\uff0c\u5219R=(1/2)\u221a3a\uff0c
2\u3001\u6b63\u56db\u9762\u4f53\u7684\u5185\u5207\u7403\u534a\u5f84\u4e3ar\uff0c\u5219\u5229\u7528\u4f53\u79ef\uff0c\u5f97\uff1a
(1/3)a�0�6=(1/3)\u00d7[4\u00d7(\u221a3/4)\u00d7(2a�0�5)]\u00d7r
\u5f97\uff1ar=[1/(2\u221a3)]a
\u5219\u534a\u5f84\u4e4b\u6bd4\u662f1\uff1a3\uff0c\u5219\u4f53\u79ef\u4e4b\u6bd4\u662f1\uff1a27
\u5982\u679c\u8bbe\u5185\u5207\u7403\u7684\u534a\u5f84\u4e3aX,\u5916\u5207\u7403\u7684\u534a\u5f84\u4e3aY,\u90a3\u4e48\u901a\u8fc7\u753b\u56fe\u53ef\u4ee5\u5f97\u77e5\uff0cY=\u6839\u53f72*X,\u7403\u4f53\u79ef\u516c\u5f0f\u4e3a\u4e2d\u6211\u4eec\u77e5\u9053\u7403\u4f53\u79ef\u4e0e\u7403\u534a\u5f84\u7684\u7acb\u65b9\u6210\u6bd4\uff0c\u6240\u4ee5\u5927\u5c0f\u7403\u7684\u4f53\u79ef\u6bd4\u4e3a4\uff1a1
半径比为1:根号3体积比为1:(根号3)^3=1:3根号3
绛旓細鍗婂緞涔嬫瘮=鐩村緞涔嬫瘮=1锛氭牴鍙3銆鍐呭垏鐞鐨勭洿寰勬槸姝ｅ洓闈綋鐨勮竟闀匡紝澶栨帴鐞冪殑鐩村緞鏄綋瀵硅绾跨殑闀垮害锛岃姝ｅ洓闈綋鐨勮竟闀夸负a锛屽垯浣撳瑙掔嚎鐨勯暱搴=锛堟牴鍙3锛塧銆傚疄鍦ㄤ笉琛屽氨寤哄潗鏍囩郴锛屽垪鍑虹偣鐨勫潗鏍囩敤鍕捐偂瀹氱悊鍋氥傝櫧璇存病鍟ョ編鎰熶絾鏄畝鍗曠矖鏆寸瀛︽湁鏁堛傝屼笖杩樺彲浠ョ鍒ゆ槸鍚︽湁澶栨帴鐞冿紝鍒瓑姹備簡鍗婂ぉ鍙戠幇鍏跺疄娌℃湁澶栨帴...
绛旓細1銆澶栨帴鐞銆傝竟闀夸负a鐨姝ｅ洓闈綋鍙互鐪嬫垚鏄竟闀挎槸锛堚垰2/2锛塧鐨勬鏂逛綋鎴嚭鏉ョ殑锛屽垯鍏跺鎺ョ悆鐩村緞鏄鏂逛綋杈归暱鐨勨垰3鍊嶃2銆佸唴鍒囩悆鍗婂緞銆傝姝ｅ洓闈綋鏄疭锛岮BC锛岃繃鐐筍浣滈珮绾縎H浜ゅ簳闈BC浜庣偣H锛屽垯鍐呭垏鐞冪悆蹇冨湪SH涓婏紝璁惧叾鍗婂緞鏄疪锛屽垯涓昏灏变骇鐢熷洓涓洓闈綋锛歄锛峉AB銆丱锛峉BC銆丱锛峉CA銆丱锛岮BC锛岃繖...
绛旓細璁剧悆鍗婂緞涓簉锛岀敱RtPEO鈭絉tPO1DEP6ra12ADBOCO13銆佽嫢姝ｅ洓浣撶殑妫遍暱閮戒负a锛屽唴鏈変竴鐞冧笌鍥涗釜闈㈤兘鐩稿垏锛姹傜悆鐨鍗婂緞瑙ｆ硶2锛氳繛缁揙A銆丱B銆丱C銆丱P锛岄偅涔圥VPABCVOPABVOPBCVOPCAVOABC4VOABCVOABC鍥燰PABC1SABCOO1,31SABCPO1,3ADEOCO1B鎵浠O14r6ra12姹傛１闀夸负a鐨姝ｅ洓闈綋鐨澶栨帴鐞冨拰瀹冪殑鍐呭垏鐞冪殑浣撶Н...
绛旓細瑙ｏ細璁姝ｅ洓闈綋涓篜ABC锛屼袱鐞冪悆蹇冮噸鍚堬紝璁句负O锛 璁綪O鐨勫欢闀跨嚎涓庡簳闈BC鐨勪氦鐐逛负D锛屽垯PD涓烘鍥涢潰浣揚ABC鐨勯珮锛孭D鈯ュ簳闈BC锛屼笖PO=R锛孫D=r锛孫D=姝ｅ洓闈綋PABC鍐呭垏鐞冪殑楂橈紟璁炬鍥涢潰浣揚ABC搴曢潰闈㈢Н涓篠锛 灏嗙悆蹇僌涓庡洓闈綋鐨4涓《鐐筆ABC鍏ㄩ儴杩炴帴锛屽彲浠ュ緱鍒4涓叏绛夌殑姝ｄ笁妫遍敟锛岀悆蹇冧负椤剁偣锛屼互姝...
绛旓細杩炴帴姝ｅ洓闈綋鐨鍚勪釜涓夎褰㈢殑涓績锛屽舰鎴愪竴涓柊鐨勬鍥涢潰浣撱傚鏄撹瘉鏄庯紝鏂版鍥涢潰浣撶殑杈归暱涓篴/3.鎴戞兂锛屾寜杩欎釜鎬濊矾鍋氫笅鍘伙紝澶ф鏄瘮杈冪畝鍗曠殑鍋氭硶銆傚師鏉鍥涢潰浣撶殑鍐呭垏鍦嗘槸鏂鍥涢潰浣撶殑澶栨帴鍦嗐傛墍浠ュ鎺ュ渾鍗婂緞R鏄唴鍒囧渾鍗婂緞r鐨3鍊嶃俁=3r,浣滃浘鍗冲彲鐭ラ亾 (3r)^2=r^2+[(2/3)脳(鏍瑰彿3)a/2]^2 =>r=a/...
绛旓細璁惧叾鍗婂緞鏄疪锛屽垯涓昏灏变骇鐢熷洓涓洓闈綋锛歄锛峉AB銆丱锛峉BC銆丱锛峉CA銆丱锛岮BC锛岃繖鍥涗釜鍥涢潰浣撶殑楂橀兘鏄鍐呭垏鐞冪殑鍗婂緞R锛屽簳闈㈤兘鏄互a涓鸿竟闀挎槸姝ｄ笁瑙掑舰锛屽埄鐢ㄧ瓑浣撶Н娉曞彲浠ユ眰鍑哄唴鍒囩悆鍗婂緞R鐨勫笺傝竟闀夸负a鐨姝ｅ洓闈綋鍙互鐪嬫垚鏄竟闀挎槸(鈭2/2)a鐨勬鏂逛綋鎴嚭鏉ョ殑锛屽垯鍏澶栨帴鐞鐩村緞鏄鏂逛綋杈归暱鐨勨垰3鍊嶃
绛旓細鏄撶煡锛姝ｅ洓闈綋鐨勫唴鍒囩悆蹇冧笌澶栨帴鐞冨績閲嶅悎锛屾槸姝ｅ洓闈綋鐨勪腑蹇冦傜敱姝ｅ洓闈綋鐨勪竴涓《鐐笰鍚戝簳闈CD浣滃瀭绾匡紝鍗充綔楂楢H銆傚垯鐞冨績O鍦ㄩ珮绾緼H涓娿傜敱浜嶰鏄唴蹇冿紝鍒板洓涓潰鐨勮窛绂荤浉绛夛紝浠庤 VA-BCD=4VO-BCD锛屾墍浠 AH=4OH=4r 浠庤屽鎺ュ渾鍗婂緞 R=AH-OH=3r 鎵浠 鍐呭垏鐞冨拰澶栨帴鐞冪殑浣撶Н姣斾负1/27 ...
绛旓細璁炬鏂逛綋杈归暱涓2,閭ｄ箞浣撳瑙掔嚎涓2鈭3,鎵浠ヤ腑蹇僌鍒版瘡涓《鐐硅窛绂讳负鈭3,杩欐槸姝ｅ洓闈綋澶栨帴鐞冪殑鍗婂緞R锛涜屾牴鎹浘涓缓绔嬬殑鍧愭爣绯,O(1,1,1),闈1BD鏂圭▼涓簒+y+z-2=0,鎵浠鍒伴潰A1BD璺濈锛沝=|1*1+1*1+1*1-2|/鈭(1+1+1)=1/鈭3.杩欐槸鍐呭垏鐞鐨勫崐寰剅锛岄偅涔坮:R=1/鈭3:鈭3=1:3銆...
绛旓細楂樻槸鈭6a/3,澶栨帴鐞鍗婂緞鈭6a/4,鍐呭垏鐞鍗婂緞鈭6a/12锛屾１鍒囩悆鍗婂緞鈭2a/4
绛旓細澶栨帴鐞,鍒4涓《鐐硅窛绂荤浉绛 浣犱綔涓嬪浘.鐩磋涓夎褰,涓鐩磋杈规槸2/3鐨勯珮,涓涓槸鍥涢潰浣撶殑楂,鏂滆竟鏄竟闀 鍦ㄩ珮涓婂彇鐞冨績O.h-R,2/3搴曡竟鐨勯珮鍜屽崐寰勬瀯鎴愮洿瑙掍笁瑙掑舰 鍙互瑙ｅ嚭鏉
