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In[47]:=

fa = (y3 - y2)/(x3 - x2) (x - x2) + y2

Out[47]=

y2 + ((x - x2) (-y2 + y3))/(-x2 + x3)

In[48]:=

fb = (y1 - y2)/(x1 - x2) (x - x2) + y2

Out[48]=

((x - x2) (y1 - y2))/(x1 - x2) + y2

In[49]:=

fc = (y3 - y1)/(x3 - x1) (x - x1) + y1

Out[49]=

y1 + ((x - x1) (-y1 + y3))/(-x1 + x3)

In[30]:=

 = Det[({{1, x1, y1}, {1, x2, y2}, {1, x3, y3}})]

Out[30]=

-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3

In[44]:=

\[CurlyPhi]1 = ((y2 - y3) x + (x3 - x2) y + (x2 y3 - x3 y2))/

Out[44]=

((-x2 + x3) y - x3 y2 + x (y2 - y3) + x2 y3)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

In[45]:=

\[CurlyPhi]2 = ((y3 - y1) x + (x1 - x3) y + (x3 y1 - x1 y3))/

Out[45]=

((x1 - x3) y + x3 y1 - x1 y3 + x (-y1 + y3))/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

In[46]:=

\[CurlyPhi]3 = ((y1 - y2) x + (x2 - x1) y + (x1 y2 - x2 y1))/

Out[46]=

((-x1 + x2) y - x2 y1 + x (y1 - y2) + x1 y2)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

In[62]:=

d\[CurlyPhi]1 = (y2 - y3)/

Out[62]=

(y2 - y3)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

In[63]:=

d\[CurlyPhi]2 = (y3 - y1)/

Out[63]=

(-y1 + y3)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

In[64]:=

d\[CurlyPhi]3 = (y1 - y2)/

Out[64]=

(y1 - y2)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)

Ui ŕϕ

In[73]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]1 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]1) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]1 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]1) d y d x]

Out[73]=

(y2 - y3)/3

In[74]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]1 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]1) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]1 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]1) d y d x]

Out[74]=

1/6 (-y1 + y2)

In[75]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]1 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]1) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]1 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]1) d y d x]

Out[75]=

(y1 - y3)/6

Uj ŕϕ

In[76]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]2 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]2) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]2 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]2) d y d x]

Out[76]=

1/6 (-y1 + y2)

In[77]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]2 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]2) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]2 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]2) d y d x]

Out[77]=

1/3 (-y1 + y3)

In[78]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]2 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]2) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]2 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]2) d y d x]

Out[78]=

1/6 (-y2 + y3)

Uk ŕϕ

In[79]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]3 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]3) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]3 d\[CurlyPhi]1 + \[CurlyPhi]1 d\[CurlyPhi]3) d y d x]

Out[79]=

(y1 - y3)/6

In[80]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]3 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]3) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]3 d\[CurlyPhi]2 + \[CurlyPhi]2 d\[CurlyPhi]3) d y d x]

Out[80]=

1/6 (-y2 + y3)

In[81]:=

Simplify[ _x2^x1  _fa^fb (\[CurlyPhi]3 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]3) d y d x +  _x1^x3  _fa^fc (\[CurlyPhi]3 d\[CurlyPhi]3 + \[CurlyPhi]3 d\[CurlyPhi]3) d y d x]

Out[81]=

(y1 - y2)/3


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