| Let A,A',B,B',C,C',D,D',X,X',Y,Y' be points on a plane such that:
(1) slope of BD relative to AC = slope of B'D' relative to A'C'
(2) AC:BD=A'C':B'D'
(3) X,X',Y,Y' lay on lines AB,A'B',CD,C'D' respectively
(4) AX:XB=CY:YD=A'X':X'B'=C'Y':Y'D'
Prove that
[1] AC:XY=A'C':X'Y'
[2] slope of XY relative to AC = slope of X'Y' relative to A'C'
If this problem is not yet well known, I'll name it as Yee's Transformation, otherwise... fine~
solution
construct points B",D",X",Y" such that AB"CD"X"Y" is similar to A'B'C'D'X'Y'
BD//B"D" & BD=B"D" ⇒ BB"D"D is a //gram
XX"//BB"//DD"//YY"
XX":BB"=YY":DD" ⇒ XX"=YY"
XX"//YY" & XX"=YY" ⇒ XX"Y"Y is a //gram
[1] AC:XY=AC:X"Y"=A'C':X'Y'
[2] XY//X"Y"//X'Y' |
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| 由井田問題激發起的假想, 得證!
Lines L1, L2 and L3 intercept respectively L4, L5 and L6 at A B C, D E F and G H I. If any 2 of the ratios AB:BC, DE:EF and GH:HI, and any 2 of the ratios AD:DG, BE:EH and CF:FI, are the same, then so are the others.
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