如圖,P為△ABC的邊BC上的任意一點(diǎn),設(shè)BC=a,
當(dāng)B1、C1分別為AB、AC的中點(diǎn)時,B1C1=
,
當(dāng)B2、C2分別為BB1、CC1的中點(diǎn)時,B2C2=
,
當(dāng)B3、C3分別為BB2、CC2的中點(diǎn)時,B3C3=
,
當(dāng)B4、C4分別為BB3、CC3的中點(diǎn)時,B4C4=
,
當(dāng)B5、C5分別為BB4、CC4的中點(diǎn)時,B5C5=________,
…
當(dāng)Bn、Cn分別為BBn-1、CCn-1的中點(diǎn)時,則BnCn=________;
設(shè)△ABC中BC邊上的高為h,則△PBnCn的面積為________(用含a、h的式子表示).

ah
分析:設(shè)AB=b,則AB
1=
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b,AB
2=(
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+
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)b=
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b,AB
3=(
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+
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+
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)b=
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b,由此可得AB
5=(
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+
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+
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+
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+
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)b=
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b,AB
n=(
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+
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+
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+…+
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)b=
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b,即
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=
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,再利用三角形相似求B
nC
n及△PB
nC
n的面積.
解答:設(shè)AB=b,∵B
5C
5∥BC,∴△AB
5C
5∽△ABC,∴
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=
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,
B
5C
5=
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•AB
5=
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•
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b=
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a,
同理可得△AB
nC
n∽△ABC,
∴
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=
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,
B
nC
n=
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•AB
n=
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•
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b=
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a,
設(shè)△AB
nC
n中B
nC
n邊上的高為h
n,則
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=
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,即h
n=
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h,
∴S
△PBnCn=
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B
nC
n•(h-h
n)=
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ah.
故答案為:
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a,
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a,
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ah.
點(diǎn)評:本題考查了三角形相似的判定與性質(zhì)的運(yùn)用.關(guān)鍵是由易到難,找出求邊長和高的一般規(guī)律.
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