在正方形ABCD中,E是CD上动点(E不与C、D重合),CD=mDE,AE的垂直平分线FP交AD AEBC与点FHG,交AB延长线与P(1)当m=2时,求FH/AH和FH/PH的值 (2)当m=3时,求证FH+PG=HG.(3)当m为何值时G是HP中点.
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![在正方形ABCD中,E是CD上动点(E不与C、D重合),CD=mDE,AE的垂直平分线FP交AD AEBC与点FHG,交AB延长线与P(1)当m=2时,求FH/AH和FH/PH的值 (2)当m=3时,求证FH+PG=HG.(3)当m为何值时G是HP中点.](/uploads/image/z/3719094-6-4.jpg?t=%E5%9C%A8%E6%AD%A3%E6%96%B9%E5%BD%A2ABCD%E4%B8%AD%2CE%E6%98%AFCD%E4%B8%8A%E5%8A%A8%E7%82%B9%EF%BC%88E%E4%B8%8D%E4%B8%8EC%E3%80%81D%E9%87%8D%E5%90%88%EF%BC%89%2CCD%3DmDE%2CAE%E7%9A%84%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86%E7%BA%BFFP%E4%BA%A4AD+AEBC%E4%B8%8E%E7%82%B9FHG%2C%E4%BA%A4AB%E5%BB%B6%E9%95%BF%E7%BA%BF%E4%B8%8EP%EF%BC%881%EF%BC%89%E5%BD%93m%3D2%E6%97%B6%2C%E6%B1%82FH%2FAH%E5%92%8CFH%2FPH%E7%9A%84%E5%80%BC+%282%29%E5%BD%93m%3D3%E6%97%B6%2C%E6%B1%82%E8%AF%81FH%2BPG%3DHG.%283%29%E5%BD%93m%E4%B8%BA%E4%BD%95%E5%80%BC%E6%97%B6G%E6%98%AFHP%E4%B8%AD%E7%82%B9.)
在正方形ABCD中,E是CD上动点(E不与C、D重合),CD=mDE,AE的垂直平分线FP交AD AEBC与点FHG,交AB延长线与P(1)当m=2时,求FH/AH和FH/PH的值 (2)当m=3时,求证FH+PG=HG.(3)当m为何值时G是HP中点.
在正方形ABCD中,E是CD上动点(E不与C、D重合),CD=mDE,AE的垂直平分线FP交AD AEBC与点FHG,交AB延长线与P
(1)当m=2时,求FH/AH和FH/PH的值 (2)当m=3时,求证FH+PG=HG.(3)当m为何值时G是HP中点.
在正方形ABCD中,E是CD上动点(E不与C、D重合),CD=mDE,AE的垂直平分线FP交AD AEBC与点FHG,交AB延长线与P(1)当m=2时,求FH/AH和FH/PH的值 (2)当m=3时,求证FH+PG=HG.(3)当m为何值时G是HP中点.
(1)
三角形AFH∽三角形AED
FH/AH=DE/AD=DE/DC=1/m =1/2
三角形AFH∽三角形PFA
FH/AH=AH/PH=1/m
FH/PH=1/m^2 =1/4
(2)
设DE=1,CD=AD=m=3
AE=√(m^2+1)=√10
AH=AE/2=√(m^2+1)/2=√10/2
FH/AH=1/m,FH=AH/3=√(m^2+1)/6=√10/6
AH/PH=1/m,PH=3AH=m√10/2=3√10/2
PF=PH+FH=[√10/6](3m+1)=[√10/6](9+1)=5√10/3
AF=√(AH^2+HF^2)=√[10(m^2+1)]/6=5/3
AP=mAF=m√[10(m^2+1)]/6=5
BP=m√[10(m^2+1)]/6-3=AP-AB=5-3=2
PG/PF=BP/AP,PG=PF*BP/AP=2√10/3
PH-FH=3√10/2-√10/6=4√10/3=2PG
FH+PG=PH-PG=HG
(3)
设FH=1,AH=m,PH=m^2,PG=GH=m^2/2,PF=m^2+1
AF=√(AH^2+HF^2)=√(m^2+1)
AP=mAF=m√(m^2+1)
AE=2AH=2m
AD/AE=AH/AF,AD=AE*AH/AF=2m^2/√(m^2+1)=AB
AP/AB=(m^2+1)/2m = PF/FG = (m^2+1)/FG
FG=2m =FH+GH=1+m^2/2
m^2-4m+2=0
m1=2+√2 >1 (舍去)
m2=2-√2