中四三角學

三角圖像從新繪畫:
http://img170.imageshack.us/img170/9529/triangle.jpg

圖中a,b,c,d,e,f代每一小三角型面積.
角AGE=x等等如圖所示.

考慮三角ACG及GCD的高同為h,而AG,GD都=6,所以兩三角型為相等,即 a + b = c ... (1)
以相同理由, e + f = d ...(2)
考慮三角FCG及GCB面積比例為3:9或1:3
故 3b = c + d ...(3)
同理 3a = e + f ... (4)
(2)&(4) => d = 3a
代入(3) => 3b = c + 3a
因(1)=> c = a + b
所以 3b = a + b + 3a
b = 2a
c = 3a
d = 3a
e + f = 3a
總面積=12a

設GC=p, EG=q
利用三角型面積方程式,考慮部分小三角型面積:
b=1/2(3)(p)sinz=2a ... (5)
c=1/2(p)(6)sinx=3a ... (6)
f=1/2(6)(q)sinx ... (7)
e=1/2(9)(q)sinz ... (8)

(5)/(6) => 3sinz/(6sinx)=2/3
sinx/sinz =3/4 ... (A)

(7)/(8) => f/e = 6sinx/9sinz
f/e = (6/9)(3/4) = 1/2

因 f + e = 3a, f = a; e = 2a

(7)/(6) => 6qsinx/6psinx = a/3a
q/p=1/3

因p+q=20; q = 5; p = 15

(5) => 1/2(3)(15)sinz=2a ... (9)
另小三角型AGF面積為1/2(6)(3)siny=a ... (10)
(9)/(10) => 45sinz/18siny=2
sinz = 4/5 siny

因sinx/sinz=3/4, sinx=3/5 siny

考慮 x + y + z = 180
z = 180 - (x+y)
sinz = sin(x+y)
sinz = sinxcosy + sinycosx
4/5siny = 3/5cosy + sinycosx
cosx = (4 - 3cosy)/5

sin^2x + cos^2x = 1
(3/5siny)^2 + [(4 - 3cosy)/5]^2 = 1
9sin^2y + 16 -24cosy + 9cos^2y = 25
-24cosy = 25 - 16 - 9(sin^2y+cos^2y) = 0
cosy = 0 => y = 90直角

AGF面積a=1/2(3)(6)=9
總面積=12a = 108

個圖好難睇-.-

佢一隻角都冇俾你???.....><"