Hope u like my process ====================== Straight line Equations are in form of ------------------------------------------------------ => a x + by + c = 0
where (1), (2),(3) are concurrent. ___________________________ For concurrency we know that =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Determinant of the a, b, c of all 3 straight lines will be 0.
For isosceles triangle.. =-=-=-=-=-=-=-=-=-=-=-=- Any of the two angles of triangle should be equal. _________________________ Here determinant is taken out by elementary row operations where I first did R1` = R1 - R2 and R2` =R2 - R3
Plz.. Follow the pic...
Sorry I though to explain here by writings but time expired. ___________________________ Hope this is ur required answer
rakeshmohata
and.. I should get much time. bcoz I would have completed it all in more 5 min.. but as time expired I went out and again I had to begu from starting..
rakeshmohata
❣️... welcome.. and any help may ask me..
Answers & Comments
Verified answer
Hope u like my process======================
Straight line Equations are in form of
------------------------------------------------------
=> a x + by + c = 0
Given:
=-=-=-=-
=> (cos²A)x + (cos A)y + 1 =0 ____(1)
=> (cos²B)x + (cosB)y + 1 =0 _____(2)
=> (cos²C)x + (cosC)y + 1 = 0 _____(3)
where (1), (2),(3) are concurrent.
___________________________
For concurrency we know that
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Determinant of the a, b, c of all 3 straight lines will be 0.
For isosceles triangle..
=-=-=-=-=-=-=-=-=-=-=-=-
Any of the two angles of triangle should be equal.
_________________________
Here determinant is taken out by elementary row operations where I first did R1` = R1 - R2 and R2` =R2 - R3
Plz.. Follow the pic...
Sorry I though to explain here by writings but time expired.
___________________________
Hope this is ur required answer
Proud to help you ☺️
Hey friend, Harish here.
Here is your answer.
Given that the three following lines are concurrent :
So as they are concurrent we can represent the following determinant.
And it is interesting to note that it is a 3 x 3 Vandermonde determinant which equals :
Hence atleast two of the cosines or equal.
So, As two angles are atleast equal the triangle formed with these angles must be a isosceles triangle.
____________________________________________________
Hope my answer is helpful to you.
Thanks @RakeshMohata for you answer. Just simplified into few steps for clear understanding.