如图,等边三角形ABC中,点D、E、F分别在边BC、CA、AB上,且BD=2DC,BE=2EC,CF=2FA,AD与BE相交于点P,BE与CF相交于点Q,CF与AD相交于点R,则AP :PR :RD=_______.若△ABC的面积为1,则△PQR的面积为______.
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![如图,等边三角形ABC中,点D、E、F分别在边BC、CA、AB上,且BD=2DC,BE=2EC,CF=2FA,AD与BE相交于点P,BE与CF相交于点Q,CF与AD相交于点R,则AP :PR :RD=_______.若△ABC的面积为1,则△PQR的面积为______.](/uploads/image/z/11683953-9-3.jpg?t=%E5%A6%82%E5%9B%BE%2C%E7%AD%89%E8%BE%B9%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%AD%2C%E7%82%B9D%E3%80%81E%E3%80%81F%E5%88%86%E5%88%AB%E5%9C%A8%E8%BE%B9BC%E3%80%81CA%E3%80%81AB%E4%B8%8A%2C%E4%B8%94BD%EF%BC%9D2DC%2CBE%EF%BC%9D2EC%2CCF%EF%BC%9D2FA%2CAD%E4%B8%8EBE%E7%9B%B8%E4%BA%A4%E4%BA%8E%E7%82%B9P%2CBE%E4%B8%8ECF%E7%9B%B8%E4%BA%A4%E4%BA%8E%E7%82%B9Q%2CCF%E4%B8%8EAD%E7%9B%B8%E4%BA%A4%E4%BA%8E%E7%82%B9R%2C%E5%88%99AP+%3APR+%3ARD%EF%BC%9D_______%EF%BC%8E%E8%8B%A5%E2%96%B3ABC%E7%9A%84%E9%9D%A2%E7%A7%AF%E4%B8%BA1%2C%E5%88%99%E2%96%B3PQR%E7%9A%84%E9%9D%A2%E7%A7%AF%E4%B8%BA______%EF%BC%8E)
如图,等边三角形ABC中,点D、E、F分别在边BC、CA、AB上,且BD=2DC,BE=2EC,CF=2FA,AD与BE相交于点P,BE与CF相交于点Q,CF与AD相交于点R,则AP :PR :RD=_______.若△ABC的面积为1,则△PQR的面积为______.
如图,等边三角形ABC中,点D、E、F分别在边BC、CA、AB上,且BD=2DC,BE=2EC,CF=2FA,AD与BE相交于点P,BE与CF相交于点Q,CF与AD相交于点R,则AP :PR :RD=_______.若△ABC的面积为1,则△PQR的面积为______.
如图,等边三角形ABC中,点D、E、F分别在边BC、CA、AB上,且BD=2DC,BE=2EC,CF=2FA,AD与BE相交于点P,BE与CF相交于点Q,CF与AD相交于点R,则AP :PR :RD=_______.若△ABC的面积为1,则△PQR的面积为______.
应该是:BD=2DC,CE=2AE,AF=2BF.
显然,图中有好几组三个一组的全等三角形,这里不再详细证明.
过点E作EM平行于BC,交AD于M,作EN平行于AB,交CF于N.则有:
△AME与△ADC相似,EM/DC=AE/AC=1/3,
△EMP与△BDP相似,EP/PB=EM/BD=EM/2DC=1/6,EP=1/7EB.
△CEN与△CAF相似,EN/AF=CE/CA=2/3,
△ENQ与△BFQ相似,EQ/QB=EN/BF=EN/(1/2AF)=4/3,QB=3/7EB.
所以AP:PR:RD=BQ:QP:EP=3:3:1.
BD=2DC,得S(△ADC)=1/3S(△ABC),
RD=1/7AD,得S(△CAR)=6/7S(△ADC)=2/7S(△ABC)=S(△ABP)=S(△BCQ)=2/7,
所以△PQR的面积为1-6/7=1/7.
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