The centroid of a triangle,whose vertices are $(2, 1)$,$(5, 2)$,and $(3, 4)$,is

Find the distance of the line $4x - y = 0$ from the point $P(4, 1)$ measured along the line making an angle of $135^{\circ}$ with the positive $x$-axis.

If $(a, b - c)$,$(b, c - a)$,and $(c, a - b)$ are the vertices of a triangle,then where does the centroid of the triangle lie?

Two vertices of a triangle $ABC$ are $A(3, -1)$ and $B(-2, 3)$,and its orthocentre is $P(1, 1)$. If the coordinates of the point $C$ are $(\alpha, \beta)$ and the centre of the circle circumscribing the triangle $PAB$ is $(h, k)$,then the value of $(\alpha + \beta) + 2(h + k)$ equals :

If the midpoints of the sides of a triangle are $(5, 0)$,$(5, 12)$,and $(0, 12)$,then what is the orthocenter of this triangle?

The orthocentre of the triangle formed by the lines $x + y = 1$,$2x + 3y = 6$,and $4x - y + 4 = 0$ lies in which quadrant?