【答案】
分析:(1)本題可利用DE∥BC,根據(jù)平行線分線段成比例定理,來(lái)求出x、y的函數(shù)關(guān)系式.
(2)本題要分兩種情況:
①兩圓外切,根據(jù)∠A的余弦值,如果過(guò)B作AC的垂線,不難得出△ABC為等腰三角形,因此AB=BC=5(也可用余弦定理求出BC的長(zhǎng)).
那么△ADE也應(yīng)該是等腰三角形,即AD=DE=5-y.
由于兩圓外切,設(shè)以BD為直徑的圓為⊙O
1,以CE為直徑的圓為⊙O
2,那么O
1O
2就是梯形DECB的中位線,根據(jù)DE、BC的長(zhǎng)即兩圓的半徑即可求出DE的長(zhǎng).
②兩圓內(nèi)切,此種情況又要分兩種情況來(lái)求:
一:⊙O
2內(nèi)切于⊙O
1,那么O
1O
2是兩圓的半徑差,可根據(jù)相似三角形ADE和AO
1O
2來(lái)求出DE的長(zhǎng).
二:⊙O
1內(nèi)切于⊙O
2,同一.
(3)本題也要分三種情況:
①當(dāng)∠ADE=∠FDE時(shí),由于DE∥BC,那么∠ADE=∠FDE=∠DFB=∠B,即AD=DF=DE=DB,如果連接AF,那么DE必垂直平分AF,因此AF⊥CB,在直角三角形AFC中,由(2)知:∠A=∠C,因此根據(jù)AC的長(zhǎng)和∠C的余弦值即可求出FC的長(zhǎng)進(jìn)而可求出BF的長(zhǎng).
②當(dāng)∠DEF=∠B時(shí),此時(shí)∠ADE=∠B=∠DEF,因此AB∥EF,四邊形BDEF為平行四邊形.因此△ADE≌△BDF,因此BF=BD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/0.png)
AB,由此可求出BF的長(zhǎng).
③當(dāng)∠DFE=∠B時(shí),可根據(jù)相似三角形對(duì)應(yīng)的腰和底成比例求出BF的長(zhǎng).
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images1.png)
解:(1)∵DE∥BC,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/1.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/2.png)
,
∴y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/3.png)
x(x>0且x≠3).
(2)作BH⊥AC,垂足為點(diǎn)H.
∵cosA=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/4.png)
,AB=5,
∴AH=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/5.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/6.png)
AC,
∴BH垂直平分AC.
∴△ABC為等腰三角形,AB=CB=5.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images8.png)
①當(dāng)點(diǎn)D在BA邊上時(shí)(兩圓外切),如圖(1)
易知:O
1O
2∥BC,∴O
1O
2=AO
1,
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/7.png)
+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/8.png)
=5-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/9.png)
.
∵y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/10.png)
x,
∴x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/11.png)
.
∵DE∥BC,
∴DE=AD=5-y,
∴DE=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/12.png)
x+5.
∴DE=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/13.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/14.png)
+5=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/15.png)
;
②當(dāng)點(diǎn)D在BA延長(zhǎng)線上時(shí)(兩圓內(nèi)切),如圖(2)、(3),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images18.png)
易知O
1O
2∥BC,且O
1O
2=AO
1,
(�。┤鐖D(2),
∵O
1O
2=AO
1,
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/16.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/17.png)
=5-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/18.png)
.
∵y=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/19.png)
x,
∴x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/20.png)
.
∵DE∥BC,
∴DE=AD=y-5,
∴DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/21.png)
x-5.
∴DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/22.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/23.png)
-5=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/24.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images28.png)
(ⅱ)如圖(3),
∵O
1O
2=AO
2,
即
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/25.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/26.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/27.png)
-5,
∴x=10.
∵DE∥BC,
∴DE=AD=y-5,
∴DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/28.png)
x-5.
∴DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/29.png)
×10-5=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/30.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images35.png)
(3)①當(dāng)∠EDF=∠B時(shí),
易得:AD=DE=DF=DB,
∴AF⊥BC,
由cosA=cosC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/31.png)
,AC=3,
∴FC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/32.png)
,∴BF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/33.png)
.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images39.png)
②當(dāng)∠DEF=∠B時(shí),如圖(5)
易得:△DBF≌△EFC,
∴BF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/34.png)
.
③當(dāng)∠DFE=∠B時(shí),如圖(6)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/35.png)
,
∵AB=5,BC=5,AC=3,
設(shè)DE=3k,DF=EF=5k,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/36.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/images43.png)
∴k=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/37.png)
,
∴BF=5-3k=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/38.png)
.
綜上所述:BF的長(zhǎng)為:BF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/39.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/40.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022165114754148591/SYS201310221651147541485024_DA/41.png)
.
點(diǎn)評(píng):本題考查了等腰三角形的判定和性質(zhì)、圓與圓的位置關(guān)系、相似三角形的判定和性質(zhì)等知識(shí).