Lines on a plane

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Problem: Given the coordinates of the vertices of the triangle \( A B C-A(1,-6), B(3,4), C(-3,3) \). Find: 1.The equation of the height of \( \mathrm{CH} \) and its length; 2. The equation of the median of \( A M \) and its length; 3.The angle formed by the height of \( \mathrm{CH} \) and the median of \( A M \).

3.2.1 Lines on a plane

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Problem: Given the equations of a side of a triangle and the bisectors of two angles, adjacent to this side. Find the equations of the other two sides of the triangle. The side: \( 4 x+3 y=0 \), bisectors: \( 7 x+4 y+5=0 \), \( y+4=0 \).

3.2.2 Lines on a plane

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Problem: On the sides \( A B \) and \( A C \) of the given triangle \( A B C \) points \( M \) and \( N \) are taken respectively, that divide the segments \( A B \) and \( A C \) with respect to \( 3: 1 \) counting from the vertex \( A \), and the point \( Q \) is the midpoint of segment \( B C \). Prove that the straight lines \( A Q, B N, C M \) intersect at one point. Given the coordinates of the vertices of the triangle \( A B C: A(-3,-1), B(1,-5), C(9,3) \).

3.2.3 Lines on a plane

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Problem: Find the equation of the straight line at the distance \( \rho \) from the point \( A \) and the component with the axis \( O X \) twice the size of the angle of the straight line \( l \), if: \[ A(-1,2), \quad \rho=\sqrt{34}, \quad l: 2 x-6 y+5=0 . \]

3.2.4 Lines on a plane

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