\(2bc+b^2+c^2-a^2\)

\(=\left(b+c\right)^2-a^2\)

\(=\left(b+c+a\right)\cdot\left(b+c-a\right)\)

\(=2p\cdot\left(2p-a-a\right)\)

\(=4p\left(p-a\right)\)

TC:a+b+cd=2p=>b+c=2p-a

=>(b+c)²=(2p-a)²

=>b²+2bc+c²=4p²-4pa+a²

=>b²+2bc+c²-a²=4p²-4pa

=>2bc+b²+c²-a²=4p(p-a) ĐPCM

\(2bc+b^2+c^2-a^2\)

\(=\left(b+c\right)^2-a^2\)

\(=\left(a+b+c\right)\left(b+c-a\right)\)

\(=2p\left(a+b+c-2a\right)\)

\(=2p\left(2p-2a\right)=4p\left(p-a\right)\)

biến đổi vế phải ta được:

4p(p -a ) = 4p\(^2\)-4pa

=(2p)\(^2\)-2p.2a

=(a+b+c)\(^2\)-2a(a+b+c)

=\(a^2+b^2+c^2+2ab+2ac+2bc\)-\(2a^2-2ab-2ac\)

=\(2bc+b^2+c^2-a^2\)=vế trái (đpcm)

\(2bc+b^2+c^2-a^2\)

\(=\left(b+c\right)^2-a^2\)

\(=\left(b+c-a\right)\left(b+c+a\right)\)

\(=\left(b+c+a-2a\right).2p\)

\(=\left(2p-2a\right).2p\)

\(=4p\left(p-a\right)\)\(\left(ĐPCM\right)\)

\(2bc+b^2+c^2-a^2=4p\left(p-a\right)\)

Biến đổi vế phải ta có :

\(4p\left(p-a\right)\)

\(=2p\left(2p-2a\right)\)

\(=\left(a+b+c\right)\left(b-c-a\right)\)

\(=2bc+b^2+c^2-a^2=VT\)(đpcm)

\(a+b+c=2p\Rightarrow\frac{a+b+c}{2}=p\Rightarrow p-a=\frac{b+c-a}{2}\Rightarrow\left(b+c-a\right)=2\left(p-a\right)\)

Và: \(2bc+b^2+c^2-a^2=\left(b+c\right)^2-a^2=\left(b+c-a\right)\left(b+c+a\right)=2\left(p-a\right)\cdot2p=4p\left(p-a\right)\)đpcm.

1, a +b +c = 0 => a + b = -c ; a +c = -b ; b+c = -a

thay vào M ta có

 M = a . -c . -b = abc (1)

Thay tương tự vào  N , P ta cũng đc N =abc (2)

                                                       P =abc( 3)

Từ 1 2 và 3 => ĐPCM

2,

a + b +c = 2P

=>  b + c = 2P -a

=> ( b + c)^2 = ( 2P -a)^2

=> b^2 + 2bc+ c^2 = 4p^2 - 4pa + a^2

=> 2bc+ b^2 + c^2 -a^ 2 = 4p^2 - 4pa

=> 2bc + b^2 + c^2 -a ^ 2 = 4p(p-a)=> ĐPCM

1.

Ta có a+b+c=0

=> a+c=b ; a+b=c ; c+b=a

M= a(a+b)(a+c)=a.c.b

N= b(b+c)(a+b)=b.a.c

P= c(c+a)(c+b)=c.b.a

=> M=N=P=abc

4p(p-a)=2(a+b+c)[(b+c-a)/2]=(a+b+c)(c+b-a)⁽¹⁾

b²+c²+2ab-a²=(a+b+c)(c+b-a)⁽²⁾

từ ⁽¹⁾ và ⁽²⁾ suy ra b²+c²+2ab-a²=4p(p-a) 

a+b+c = 2p => 4p = 2(a+b+c); p=(a+b+c)/2

VP = 4p(p-a) = 2(a+b+c)(\(\frac{a+b+c}{2}-a\))

= \(2\left(a+b+c\right)\left(\frac{a+b+c-2a}{2}\right)\)

=\(2\left(a+b+c\right)\cdot\frac{b+c-a}{2}=\left(a+b+c\right)\left(b+c-a\right)\)

\(=ab+ac-a^2+b^2+bc-ab+bc+c^2-ac\)

\(=2bc+b^2+c^2-a^2\) = VT (đpcm)