Maths Question

設ABC邊長為Z
因APB : BPC : CPA = 5:6:7
APB = 100; BPC = 120; CPA = 140
設ACP = X
則PCB = 60 - X [ABC為等邊三角形]
PBC = 180 - 120 - (60 - X) = X
ABP = 60 - X
考慮三角形PBC, Z/sin120 = PC/sinX = PB/sin(60-X)
PC = ZsinX/sin120...(1)
PB = Zsin(60-X)/sin120...(2)
考慮三角形PCA, Z/sin140 = PA/sinX
PA = ZsinX/sin140 ... (3)
考慮三角形PAB, PA/sin(60-X) = Z/sin100
PA = Zsin(60-X)/sin100 ... (4)
(2)/(4) => PB/PA = sin100/sin120
PB/PA = sin80/sin60 因sin(180 - A) = sinA
PB/sin80 = PA/sin60 ... (5)
(1)/(3) => PC/PA = sin140/sin120
PC/PA = sin40/sin60
PC/sin40 = PA/sin60 ...(6)
(5),(6) => PA/sin60 = PB/sin80 = PC/sin40
60 + 80 + 40 = 180 = 三角內角和
所以以PA,PB,PC為邊長的三角形其內角分別為60,80及40
比例為3:4:2

圖片參考：http://img133.imageshack.us/img133/349/tri.png

http://img133.imageshack.us/img133/349/tri.png