如图，已知平行四边形ABCD中，E、F分别为边AB、AD上的点．EF与对角线AC交于P，若AEEB＝ab，AFFD＝mn（a、 平行四边形ABCD中E、F分别是AB、AD上的点，EF与对角...
\u5982\u56fe\uff0c\u5df2\u77e5\u5e73\u884c\u56db\u8fb9\u5f62ABCD\u4e2d\uff0cE\u3001F\u5206\u522b\u4e3a\u8fb9AB\u3001AD\u4e0a\u7684\u70b9\uff0cEF\u4e0e\u5bf9\u89d2\u7ebfAC\u4ea4\u4e8e\u70b9P\uff0e\u82e5AEEB=ab\uff0cAFFD=mn\uff08a\u3001\u8fc7\u70b9E\u4f5cEG\u2225AD\uff0c\u4ea4AC\u4e8e\u70b9O\uff0c\u2235\u56db\u8fb9\u5f62ABCD\u662f\u5e73\u884c\u56db\u8fb9\u5f62\uff0c\u2234AD\u2225EG\u2225BC\uff0cAD=BC\uff0c\u2234AOOC\uff1dAEBE\uff1dab\uff0c\u25b3AEO\u223d\u25b3ABC\uff0c\u25b3APF\u223d\u25b3OPE\uff0c\u2234AP+POPC?PO\uff1dab\uff0cEOBC\uff1dAEAB\uff1daa+b\uff0cAFEO\uff1dAPPO\uff0c\u2235AEEB\uff1dab\uff0cAFFD\uff1dmn\u2234\u4ee4AE=ax\uff0cBE=bx\uff0cAF=my\uff0cDF=ny\uff0c\u2234EOmy+ny\uff1daa+b\uff0c\u2234EO=ay(m+n)a+b\uff0c\u2234myay(m+n)a+b\uff1dAPPO\uff0c\u2234AP\uff08a+b\uff09bm+AP\uff08m+n\uff09ab+AP\uff08m+n\uff09a2=PC\uff08a+b\uff09am\uff0c\u2234AP\uff08bm+an+am\uff09\uff08a+b\uff09=PC\uff08a+b\uff09am\uff0c\u2234APPC\uff1damam+an+bm\uff0c\u2234C\u7b54\u6848\u6b63\u786e\uff0c\u6545\u9009C\uff0e
\u89e3\uff1a\u8fc7\u70b9E\u4f5cEG\u2225AD\uff0c\u4ea4AC\u4e8e\u70b9O\uff0c
\u2235\u56db\u8fb9\u5f62ABCD\u662f\u5e73\u884c\u56db\u8fb9\u5f62\uff0c
\u2234AD\u2225EG\u2225BC\uff0cAD=BC\uff0c
\u2234AOOC=AEBE=ab\uff0c\u25b3AEO\u223d\u25b3ABC\uff0c\u25b3APF\u223d\u25b3OPE\uff0c
\u2234AP+POPC-PO=ab\uff0cEOBC=AEAB=aa+b\uff0cAFEO=APPO\uff0c
\u2235AEEB=ab\uff0cAFFD=mn
\u2234\u4ee4AE=ax\uff0cBE=bx\uff0cAF=my\uff0cDF=ny\uff0c
\u2234EOmy+ny=aa+b\uff0c
\u2234EO=ay(m+n)a+b\uff0c
\u2234myay(m+n)a+b=APPO\uff0c
\u2234PO=AP•(M+N)a(a+b)m\uff0c
\u2234AP+AP•(M+N)a(a+b)mPC-AP•(M+N)a(a+b)m=ab\uff0c
\u2234AP\uff08a+b\uff09bm+AP\uff08m+n\uff09ab+AP\uff08m+n\uff09a2=PC\uff08a+b\uff09am\uff0c
\u2234AP\uff08bm+an+am\uff09=PC\uff08a+b\uff09am\uff0c
\u2234APPC=amam+an+bm\uff0c
\u2234C\u7b54\u6848\u6b63\u786e\uff0c
\u6545\u9009C\uff0e
\u53c2\u8003\uff1ahttp://www.jyeoo.com/math/ques/detail/c3643f37-d6e7-48cd-ac1a-2eb688cb4882
∵四边形ABCD是平行四边形，
∴AD∥EG∥BC，AD=BC，
∴
| AO |
| OC |
| AE |
| BE |
| a |
| b |
∴
| AP+PO |
| PC?PO |
| a |
| b |
| EO |
| BC |
| AE |
| AB |
| a |
| a+b |
| AF |
| EO |
| AP |
| PO |
∵
| AE |
| EB |
| a |
| b |
| AF |
| FD |
| m |
| n |
∴令AE=ax，BE=bx，AF=my，DF=ny，
∴
| EO |
| my+ny |
| a |
| a+b |
∴EO=
| ay(m+n) |
| a+b |
∴
| my | ||
|
| AP |
| PO |
∴PO=
| AP?(M+N)a |
| (a+b)m |
∴
AP+
| ||
PC?
|
| a |
| b |
∴AP（a+b）bm+AP（m+n）ab+AP（m+n）a2=PC（a+b）am，
∴AP（bm+an+am）（a+b）=PC（a+b）am，
∴
| AP |
| PC |
| am |
| am+an+bm |
∴C答案正确，
故选C．
绛旓細鈭鍥涜竟褰BCD鏄骞宠鍥涜竟褰 鈭碅D=BC锛屽張鈭礎E=CF 鈭碅D-AE=BC-CF 鍗矰E=BF 鈭礑E鈭F 鈭村洓杈瑰舰BEDF鏄钩琛屽洓杈瑰舰 鈭碆E=DF 鈭碝銆丯鍒嗗埆鏄疊E銆丏F鐨勪腑鐐 鈭碋M=BE/2=DF/2=NF 鑰孍M鈭F 鈭村洓杈瑰舰MFNE鏄钩琛屽洓杈瑰舰
绛旓細锛1锛塃G=FH锛庯紙2锛夎瘉鏄庯細鈭鍥涜竟褰BCD鏄骞宠鍥涜竟褰,鈭碅D鈭C,鈭粹垹BAD+鈭燗BC=180掳锛庡張鈭礎F,BH鍒嗗埆骞冲垎鈭燘AD,鈭燗BC,鈭粹垹DAE=鈭燗EB=12鈭燚AB,鈭粹垹BAE+鈭燗BE=90掳,鈭粹垹AEB=90掳,鈭粹垹FEH=90掳锛庡悓鐞嗗彲璇佲垹EFG=90掳,鈭燛HG=90掳,鈭村洓杈瑰舰EFGH涓虹煩褰,鈭碋G=FH锛
绛旓細璇佹槑锛氣埖ABCD鏄骞宠鍥涜竟褰 鈭碅B=CD锛孉B//CD 鈭粹垹BAE =鈭燚CF 鍙堚埖AE =CF 鈭粹娍ABF鈮屸娍CDF锛圫AS锛夆埓BE=DF 鈭燗EF=鈭燙FD 鈭粹垹BEC=鈭燚FA銆愬墠涓や釜瑙掔殑琛ヨ銆戔埓BE//DF銆愬唴閿欒鐩哥瓑銆
绛旓細鈭骞宠鍥涜竟褰BCD 鈭碅B锛滳D 鈭礏A锛滲E 鈭碈E锛滲C-BE锛滲C-AB 鈭礏C锛2AB 鈭碈E锛2AB-AB锛滱B 鈭碈E锛滳D 鈭粹垹CED锛濓紙180-鈭燚锛/2 鍙堚埖BE锛滲A 鈭粹垹BEA锛濓紙180-鈭燘锛/2 鈭粹垹AEF锛180-鈭燘EA-鈭燙ED 锛180-锛180-鈭燘锛/2-锛180-鈭燚锛/2 锛濓紙鈭燘+鈭燚锛/2 鈭礎B鈭D 鈭粹垹B+鈭燚锛...
绛旓細浣犲ソ锛屼繆鐙肩寧鑻辩湡璇氫负浣犺В绛旓紝鏈纭湁闅惧害锛屽氨鏄庢牱宸茬煡鏁版嵁锛屾楠や篃澶嶆潅銆傝繃M浣淢E鈭D浜D鐨勫欢闀跨嚎浜嶦锛屸埖AE鈭C锛屸埓鍥涜竟褰MED鏄骞宠鍥涜竟褰紝鈭碊E=MB=1/2BC=5锛孧E=BD=12 鈭碅E=15锛孉E^2=225锛孉M^2=81锛孧E^2=144 鈭礎E^2=AM^2+ME^2锛屸埓鈭燗ME=90掳銆傗埓AM鈯D銆傚彟涓鏂归潰锛岃AM...
绛旓細锛1锛夆埖鍥涜竟褰BCD涓骞宠鍥涜竟褰 鈭碆D骞冲垎AC 鈭碠A=OC鍙 鈭礒銆丗涓篛A銆丱C涓偣 鈭碠E=OF 锛2锛夊簲璇ユ槸BE//DF鍚 鈭礎BCD涓哄钩琛屽洓杈瑰舰 鈭碆O=OD 鍙堚埖OE=OF,鈭燚OF=鈭燛OB 鈭粹柍EOB鈮屸柍FOD锛堣竟瑙掕竟锛夆埓鈭燘EO=鈭燨FD 鈭碆E//FD
绛旓細璇佹槑锛氣埖骞宠鍥涜竟褰BCD锛CD鈭B锛孉D鈭C锛屸埖EF鈭C锛屸埓EF鈭C鈭D锛屸埓鍥涜竟褰DFE銆丆FEB鏄钩琛屽洓杈瑰舰锛屸埓FM=AM锛孎N=BN锛屾墍浠N=1/2AB
绛旓細鍥犱负E,F鏄疉D,BC涓偣 AE=DE,BF=CF 鍥犱负鍥涜竟褰BCD鏄骞宠鍥涜竟褰 AD=鈭モ懘BC 鎵浠E=FC(=寰) 鈭燛AF=鈭燘FA(鈭モ懘寰)鎵浠ュ洓杈瑰舰AFCE鏄钩琛屽洓杈瑰舰 鎵浠F鈭C鈶 鈭燗FB=鈭燛CB 鎵浠モ垹EAF=鈭燛CB 鍚岀悊鈭燘GF=鈭燜HC 鍦ㄤ笁瑙掑舰AGE,CHF涓 EAG=FCH(瑙)AGE=CHF AE=CF AGE鈮孋HF EG=FH 鍚岀悊DEH鈮孊GF...
绛旓細璇佹槑锛氣埖骞宠鍥涜竟褰BCD 鈭碅D锛滲C锛屸垹A锛濃垹C 鈭礎E锛滳F 鈭粹柍ADE鍏ㄧ瓑浜庘柍CBF 锛圫AS锛夆埓DE锛滲F
绛旓細鈶粹埖EF鈭D,鈭碈F/CD=EF/BD=8/12=2/3,鈭碊F/CD=1/3,鈭ABCD鏄骞宠鍥涜竟褰,鈭碅B+CD,鈭碊F锛欰B=1锛3.鈶碘埖AB鈭D,鈭碏H/AH=DH/DF/AB=1/3,鈭碅H/AF=3/4,鈭碐H/EF=AH/AF=3/4,F+GH=3/4脳8=6
