There are three concurrent lines who line equations are :
$$ \big( \cos^2 \mathrm A \big )x + \big (\cos \mathrm A \big )y+1 = 0$$
$$ \big( \cos^2 \mathrm B \big ) x+ \big ( \cos \mathrm B \big )y+1 = 0$$
$$ \big (\cos^2 \mathrm C \big ) x+ \big ( \cos \mathrm C\big )y +1= 0$$
It is also given that A,B,C are angles of a triangle. We should prove that the triangle is isosceles.
I understood that by saying that these three are concurrent , all three equations have a common solution. But my problem is , i am not able to find that common solution. Please tell me some idea for finding it and also tell me whether it is required to solve this question.
The compatibility condition for this system is $$\begin{vmatrix}\cos^2A&\cos A&1\\\cos^2B&\cos B&1\\\cos^2C&\cos C&1\\\end{vmatrix}=0.$$
We recognize a $3\times3$ Vandermonde determinant, which equals $$(\cos A-\cos B)(\cos A-\cos C)(\cos B-\cos C).$$
Hence the cosines of at least two angles are equal.