a-maths~~~~~~~~

(a)(i)(1) OC = λa,
OE = [h*OC + OB] / (1+h) = hλ/(1+h) a + 1/(1+h) b

(2) OD = γb,
OE = [k*OD + OA] / (1+k) = 1/(1+k) a + kγ/(1+k) b

(ii) Since hλ/(1+h) a + 1/(1+h) b = 1/(1+k) a + kγ/(1+k) b
thus hλ/(1+h)=1/(1+k) and 1/(1+h)=kγ/(1+k)
hence, λ=(1+h)/[h(1+k)] andγ=(1+k)/[k(1+h)]

(iii) OF = [|FB|*OA + |AF|*OB] / |AB|
and since OE//OF, thus
|AF| / |FB| = 1/(hλ) = kγ = (1+k)/(1+h) = [γ(1-λ)]/[λ(1-γ)]
The last equality can be varify by using (a)(ii).

(b) Let QU:UT=k:1 and RU:US=h:1
by (a)(ii), we have λ=(1+h)/[h(1+k)]=3/5 and γ=(1+k)/[k(1+h)]=4/7

If PU join QR at V, then by (a)(iii),
|QV| / |VR| = 1/(hλ) = kγ = (1+k)/(1+h) = [γ(1-λ)]/[λ(1-γ)]
5/(3h) = 4k/7 = [4/7*2/5] / [3/5*3/7] = 8/9

Thus h = 15/8, k = 14/9
i.e. QU:UT=14:9 and RU:US=15:8

2009-10-07 20:34:03 補充：
太耐冇上黎做數，希望冇做錯。