a^2+b^2=BP^2+PD^2a+b=BP+PD这两道方程联立成方程组之后,该怎么解出BP和PD?

Question

a^2+b^2=BP^2+PD^2
a+b=BP+PD
这两道方程联立成方程组之后,该怎么解出BP和PD?

Answer 1

a^2+b^2=BP^2+PD^2 ①
a+b=BP+PD ②
a+b=BP+PD
(a+b)^2=a^2+b^2+2ab=(BP+PD)^2=BP^2+PD^2+2BP*PD ③
③-① 2ab=2BP*PD
ab=BP*PD ④
②和④联立
a+b=BP+PD
ab=BP*PD
由a+b=BP+PD 得a=BP+PD -b
代入得 (BP+PD -b)*b=BP*PD
b^2-(BP+PD）*b+BP*PD =0
（b-BP)*(b-PD)=0
b=BP or b=PD
分别代入a+b=BP+PD 得
b=BP,a=PD
b=PD,a=BP