Exercises: Notable lines and 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 the vertex $A$ in the triangle with vertices $A(0, 0)$, $B(4, -2)$ and $C(2, 6)$.

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

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

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

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

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

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 trees in an agricultural plantation are located at the points $A(1, 1)$, $B(5, 3)$ and $C(1, 5)$ (coordinates expressed in tens of feet). A sprinkler is to be installed that is equidistant from the three trees. Calculate the coordinates of the point where it should be placed (circumcenter) and the water range radius necessary to water them.

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

In a triangle with vertices $A(1, 2)$, $B(3, 4)$ and $C(a, b)$, it is known that the centroid is the point $G(2, 1)$. Calculate the coordinates of the vertex $C$ and the Cartesian equation of the median starting from this vertex.

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 orthocenter of this triangle.

The orthocenter of a triangle is the point $H(2, 2)$. Two of its vertices are known, which are $A(-1, 3)$ and $B(4, 1)$. Calculate the coordinates of the vertex $C$.

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

A triangle has vertices $A(1, 1)$ and $B(5, 1)$. The vertex $C$ lies on the line of equation $y = x+2$. Knowing that the triangle is isosceles and that the unequal side is $AB$, find the coordinates of the vertex $C$ and of its circumcenter.

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

Given a generic right triangle whose vertices are located at $A(0,0)$, $B(a,0)$ and $C(0,b)$ with $a, b > 0$, prove analytically that its orthocenter coincides with the origin and its circumcenter is exactly the midpoint of the hypotenuse.

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

Given a triangle $ABC$, the vertex $A(1, 3)$ and the equations of two of its medians are known: $x-2y+1=0$ and $y-1=0$. Calculate the coordinates of the vertices $B$ and $C$.

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

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

In a triangle $ABC$, the vertex $A$ has coordinates $(-1, 2)$. The altitudes drawn from the vertices $B$ and $C$ are contained in the lines of equations $x-y+1=0$ and $2x+y-4=0$ respectively. Determine the coordinates of the vertices $B$ and $C$.

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

Find the coordinates of the vertices $B$ and $C$ of a triangle knowing that the vertex $A$ is $(3, 1)$, its circumcenter is the point $O(5, 2)$ and its centroid is $G(4, 2)$.