Lines and notable points of a triangle

Calculate the coordinates of the centroid of the triangle whose vertices are $A(1, 2)$, $B(3, -4)$ and $C(-1, 5)$.

Find the explicit equation of the median starting from vertex $A$ in the triangle with vertices $A(0, 0)$, $B(4, -2)$ and $C(2, 6)$.

Verify analytically whether the point $P(1, 2)$ is the centroid of the triangle with vertices $A(2, 3)$, $B(-1, 4)$ and $C(5, 1)$.

Determine the general equation of the altitude drawn from vertex $B$ in the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(1, -5)$.

Find the coordinates of the orthocentre of the right-angled triangle with vertices $A(0, 0)$, $B(6, 0)$ and $C(0, 4)$.

A building has a triangular floor plan with vertices at the points $A(-1, -1)$, $B(5, -1)$ and $C(2, 5)$. Calculate the coordinates of the orthocentre of this triangle.

Calculate the general equation of the perpendicular bisector of side $AB$ of a triangle, knowing that its corresponding vertices are $A(2, 5)$ and $B(-4, 1)$.

Calculate the coordinates of the circumcentre of the triangle with vertices $A(0, 0)$, $B(8, 0)$ and $C(0, 6)$.

Three trees on an agricultural plantation are located at the points $A(1, 1)$, $B(5, 3)$ and $C(1, 5)$ (coordinates expressed in tens of metres). A sprinkler is to be installed equidistant from all three trees. Calculate the coordinates of the point where it should be placed (circumcentre) and the radius of reach required to water them.

Three mountain shelters are located at the coordinates $A(0, 4)$, $B(4, 0)$ and $C(-2, -2)$ (expressed in kilometres). A rescue helicopter hovers at the circumcentre of the triangle formed by the shelters. Calculate the coordinates of the helicopter and the radius of the circle that encompasses the three shelters.

Vertex $A(2, 5)$ of a triangle $ABC$ is known. The perpendicular bisectors of sides $AB$ and $AC$ have equations $x-y+2=0$ and $2x+y-3=0$, respectively. Find the coordinates of vertices $B$ and $C$.

Calculate the equations of the two angle bisectors of the angles formed by the lines containing the sides of a triangle, whose equations are $3x-4y+1=0$ and $4x+3y-5=0$.

Three roads intersect forming a triangle bounded by the lines $x=0$, $y=0$ and $3x+4y-12=0$. Calculate the coordinates of the incentre to place a roundabout tangent to the three roads, and the radius of this roundabout.

Three telecommunications antennas form a triangle with vertices at $A(-4, 0)$, $B(2, -2)$ and $C(5, 5)$. Calculate the location of the centre of gravity (centroid) to place a central transmitter, and determine the exact distance from this transmitter to antenna $C$.

Given the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 6)$, find the coordinates of its centroid and its orthocentre. Next, obtain the general equation of the Euler line passing through both points.

The vertices of a triangle are $A(1, 0)$, $B(3, 0)$ and $C(t, 2t)$, where $t$ is a real parameter determining the position of vertex $C$ on a line. Find the equation of the locus described by the centroid of the triangle as $t$ varies.

Given the vertices of a triangle $A(-2, -1)$, $B(4, -1)$ and $C(1, 5)$, find the coordinates of the orthocentre $H$, the centroid $G$ and the circumcentre $O$. Prove analytically that the distance between the orthocentre and the centroid is twice the distance between the centroid and the circumcentre ($HG = 2GO$).