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The sides of triangle are Ur = Xcos Ar + ysinAr – Pr =0 , (R = 1,2,3) . Show that orthocentre is given by U1 cos(A2 – A3) = U2cos(A3 – A1) = U3cos(A1 – A2)

The sides of triangle are Ur = Xcos Ar + ysinAr – Pr =0 , (R = 1,2,3) . Show that orthocentre is given by U1 cos(A2 – A3) = U2cos(A3 – A1) = U3cos(A1 – A2)
                                      

Grade:11

1 Answers

Arun
25750 Points
3 years ago
Any line through the intersection of lines L 
1
​ 
  and L 
2
​ 
  and perpendicular to L 
3
​ 
  is  
cosα 
1
​ 
 cosα 
3
​ 
 +sinα 
1
​ 
 sinα 
3
​ 
 
L 
1
​ 
 
​ 
 = 
cosα 
2
​ 
 cosα 
3
​ 
 +sinα 
2
​ 
 sinα 
3
​ 
 
L 
2
​ 
 
​ 
 
L 
1
​ 
 cos(α 
2
​ 
 −α 
3
​ 
 )=L 
2
​ 
 cos(α 
3
​ 
 −α 
1
​ 
 )=L 
3
​ 
 cos(α 
1
​ 
 −α 
2
​ 
 ) by symmetry

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