證明任意四邊形的中點連起來是平行四邊形 (不用 vector)

設 A(a1, a2), B(b1, b2), C(c1, c2), D(d1, d2) 是任意四邊形的頂點
座標M,N,P,Q分別為
M = [(a1+b1)/2, (a2 +b2)/2]
N = [(b1+c1)/2, (b2 +c2)/2]
P = [(c1+d1)/2, (c2 +d2)/2]
Q = [(d1+a1)/2, (d2 +a2)/2]

slope of MN= [(a2 +b2)/2 -(b2 +c2)/2]/[(a1 +b1)/2 -(b1 +c1)/2]
=[(a2 -c2)/2]/[(a1 -c1)/2]
=(a2 -c2)/(a1 -c1)

slope of QP= [(d2 +a2)/2 -(c2 +d2)/2]/[(d1 +a1)/2 -(c1+d1)/2]
= [(a2 -c2 )/2]/[( a1 -c1)/2]
=(a2 -c2)/(a1 -c1)
=slope of MN
所以 QP//MN

slope of NP= [(b2 +c2)/2 -(c2 +d2)/2]/[(b1+c1)/2 -(c1+d1)/2]
= [(b2 -d2)/2]/[(b1-d1)/2]
= (b2 -d2)(b1-d1)

slope of MQ= [ (a2 +b2)/2 -(d2 +a2)/2]/[(a1 +b1)/2 -(d1 +a1)/2]
= [ (b2 -d2 )/2]/[(b1 -d1)/2]
= (b2 -d2)(b1-d1)
=slope of NP
所以 NP//MQ
=>MNPQ 是一個平行四邊形

2007-09-16 23:05:58 補充：
據我記憶, f.3 除左 vector 之外, point slope form, mid-point, 等等的公式都有教過. 如果夾硬唔用公式, 就唯有囉張 graph paper 黎 plot. 是但畫個四邊形, 度左幾個 mid-point, 跟住聯埋佢, 再用間尺度一度,睇下對邊係唔係平行囉. 不過我估哩個方法, 俾小學生做就正路D.

如果係數學既應該係AB//DC AD//BC而對邊對角相等,,