ABC

AB

BC

CA

L

N

E

LE

BC

H

LN

AC

J

H

J

N

E

AB

O

P

EJ

NH

HJNE

ABOP

COP

u^{2}

v^{2}

4uv

BC=a

CA=b

AB=c

\angle ACB=\gamma

p

ABC

AL=AE=p-a,~BL=BN=p-b,~CN=CE=p-c.

c\gt b

c\gt a

ABC

LH

1=\frac{CH}{HB}\cdot\frac{BL}{LA}\cdot\frac{AE}{EC}=\frac{CH}{CH+a}\cdot\frac{p-b}{p-a}\cdot\frac{p-a}{p-c}=\frac{CH}{CH+a}\cdot\frac{p-b}{p-c},

HC=\frac{a(p-c)}{c-b}

JC=\frac{b(p-c)}{c-a}

NH=NC+HC=(p-c)+\frac{a(p-c)}{c-b}=(p-c)\left(1+\frac{a}{c-b}\right)=

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

EJ=\frac{2(p-a)(p-c)}{c-a}

S_{HJNE}=\frac{1}{2}NH\cdot JE\sin\gamma=\frac{1}{2}\cdot\frac{2(p-b)(p-c)}{c-b}\cdot\frac{2(p-a)(p-c)}{c-a}\cdot\sin\gamma=

=\frac{2(p-a)(p-b)(p-c)^{2}\sin\gamma}{(c-b)(c-a)}.

CP=NP-NC=\frac{1}{2}NH-NC=\frac{(p-b)(p-c)}{c-b}-(p-c)=

=(p-c)\left(\frac{p-c}{c-b}-1\right)=\frac{(p-c)(p-b-c+b)}{c-b}=\frac{(p-c)^{2}}{c-b}

CO=\frac{(p-c)^{2}}{c-a}

v^{2}=S_{\triangle COP}=\frac{1}{2}CP\cdot CO\sin\gamma=\frac{(p-c)^{4}\sin\gamma}{2(c-b)(c-a)}.

AO=AC+CO=b+\frac{(p-c)^{2}}{c-a}=\frac{bc-ab+p^{2}-2pc+c^{2}}{c-a}=

=\frac{c(b+c-2pa)-ab-p^{2}}{c-a}=\frac{c(b+c-a-b-c)-ab-p^{2}}{c-a}=

=\frac{p^{2}-ac-ab}{c-a}=\frac{p^{2}-a(b+c)}{c-a}=\frac{p^{2}-a(2p-a)}{c-a}=

=\frac{p^{2}-2ap+a^{2})}{c-a}=\frac{(p-a)^{2}}{c-a}

BP=\frac{(p-b)^{2}}{c-b}

u^{2}=S_{ABOP}=\frac{1}{2}AO\cdot BP=\frac{1}{2}\cdot\frac{(p-a)^{2}}{c-a}\cdot\frac{(p-b)^{2}}{c-b}=\frac{(p-a)^{2}(p-b)^{2}\sin\gamma}{2(c-a)(c-b)}.

S^{2}_{HJNE}=\left(\frac{2(p-a)(p-b)(p-c)^{2}\sin\gamma}{(c-b)(c-a)}\right)^{2}=

=4\cdot2\cdot2\cdot\frac{(p-c)^{4}\sin\gamma}{2(c-b)(c-a)}\cdot\frac{(p-a)^{2}(p-b)^{2}\sin\gamma}{2(c-b)^{2}(c-a)^{2}}=16v^{2}u^{2}.

S_{HJNE}=4uv