Lines and Notable Points of a Triangle
Three telecommunications antennas are located at different places in a province, and a transmitter must be placed at the same distance from all three. Where does it go? Every triangle has special points: the center of gravity, the center of the circle that passes through its vertices, and the point where its altitudes cross. All of them are calculated with the line equations you already know.
Medians and the centroid
The formula is the average of the coordinates: the centroid is the balance point if you place the same weight at each vertex.
A line can also be written with a point and a direction vector $(a, b)$. This is the symmetric equation (some textbooks also call it the continuous equation):
$$\frac{x-x_0}{a} = \frac{y-y_0}{b}$$For a median, the point is the vertex and the direction vector is $\overrightarrow{AM}$ (from $A$ to the midpoint $M$). This equation is equivalent to $b(x-x_0) = a(y-y_0)$.
Calculate the coordinates of the centroid of the triangle whose vertices are $A(1, 2)$, $B(3, -4)$ and $C(-1, 5)$.
Calculate the coordinates of the centroid of the triangle whose vertices are $A(1, 2)$, $B(3, -4)$ and $C(-1, 5)$.
Verify analytically whether the point $P(1, 2)$ is the centroid of the triangle with vertices $A(2, 3)$, $B(-1, 4)$ and $C(5, 1)$.
Verify analytically whether the point $P(1, 2)$ is the centroid of the triangle with vertices $A(2, 3)$, $B(-1, 4)$ and $C(5, 1)$.
Find the explicit equation of the median starting from vertex $A$ in the triangle with vertices $A(0, 0)$, $B(4, -2)$ and $C(2, 6)$.
Find the explicit equation of the median starting from vertex $A$ in the triangle with vertices $A(0, 0)$, $B(4, -2)$ and $C(2, 6)$.
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 vertex $C$ and the…
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 vertex $C$ and the symmetric equation of the median starting from that vertex.
Altitudes and the orthocenter
Find the coordinates of the orthocentre of the right-angled triangle with vertices $A(0, 0)$, $B(6, 0)$ and $C(0, 4)$.
Find the coordinates of the orthocentre of the right-angled triangle with vertices $A(0, 0)$, $B(6, 0)$ and $C(0, 4)$.
Determine the general equation of the altitude drawn from vertex $B$ in the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(1, -5)$.
Determine the general equation of the altitude drawn from vertex $B$ in the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(1, -5)$.
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.
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.
The orthocentre of a triangle is the point $H(2, 2)$. Two of its vertices are known to be $A(-1, 3)$ and $B(4, 1)$. Calculate the coordinates of vertex $C$.
The orthocentre of a triangle is the point $H(2, 2)$. Two of its vertices are known to be $A(-1, 3)$ and $B(4, 1)$. Calculate the coordinates of vertex $C$.
Perpendicular bisectors and the circumcenter
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 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 $C(0, 6)$.
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 meters). A sprinkler is to be…
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 meters). A sprinkler is to be installed equidistant from all three trees. Calculate the coordinates of the point where it should be placed (circumcenter) 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 kilometers). A rescue helicopter hovers at the circumcenter of…
Three mountain shelters are located at the coordinates $A(0, 4)$, $B(4, 0)$ and $C(-2, -2)$ (expressed in kilometers). 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.
A triangle has vertices $A(1, 1)$ and $B(5, 1)$. Vertex $C$ lies on the line with equation $y = x+2$. Knowing that the triangle is isosceles and that the unequal side is…
A triangle has vertices $A(1, 1)$ and $B(5, 1)$. Vertex $C$ lies on the line with equation $y = x+2$. Knowing that the triangle is isosceles and that the unequal side is $AB$, find the coordinates of vertex $C$ and its circumcentre.
Back to the beginning: the transmitter that is equidistant from the three antennas goes at the circumcenter of the triangle they form.
Problems with unknown points
When vertices are missing, you set up the conditions of each notable line as equations and solve the system. Let's look at the most common cases.
Vertices from two medians
If $A$ is on neither of the two given medians, they both start from $B$ and $C$. Two ideas are used: the medians meet at $G$, and the midpoint $M$ of $BC$ satisfies $\overrightarrow{AM}=\tfrac32\,\overrightarrow{AG}$ (because $G$ divides $AM$ in the ratio $2:1$).
Vertices from perpendicular bisectors
The perpendicular bisector of $AB$ is the line perpendicular to $AB$ through its midpoint. So $B$ is the reflection of $A$ across that perpendicular bisector. To reflect a point $P$ across a line: draw the perpendicular to the line through $P$, find the foot $M$ (the midpoint between $P$ and its reflection), and calculate $P' = 2M - P$.
Locus: eliminating a parameter
If a point depends on a parameter $t$, its coordinates are $x = f(t)$, $y = g(t)$. To find the equation of the path it traces, you solve for $t$ in one equation and substitute into the other.
Given a generic right-angled triangle whose vertices are situated at $A(0,0)$, $B(a,0)$ and $C(0,b)$ with $a, b > 0$, prove analytically that its orthocentre coincides…
Given a generic right-angled triangle whose vertices are situated at $A(0,0)$, $B(a,0)$ and $C(0,b)$ with $a, b > 0$, prove analytically that its orthocentre coincides with the origin and its circumcentre 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 determining the position of vertex $C$ on a line. Find the equation of…
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.
For a triangle $ABC$, 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 vertices $B$ and $C$.
For a triangle $ABC$, 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 vertices $B$ and $C$.
In a triangle $ABC$, vertex $A$ has coordinates $(-1, 2)$. The altitudes drawn from vertices $B$ and $C$ lie on the lines with equations $x-y+1=0$ and $2x+y-4=0$…
In a triangle $ABC$, vertex $A$ has coordinates $(-1, 2)$. The altitudes drawn from vertices $B$ and $C$ lie on the lines with equations $x-y+1=0$ and $2x+y-4=0$ respectively. Determine the coordinates of vertices $B$ and $C$.
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…
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 $C$.
Extension: angle bisectors and the incenter
The angle bisector of an angle consists of the points that are equidistant from the two sides that form it. The three angle bisectors meet at the incenter $I$, the center of the inscribed circle (tangent to all three sides). Its radius is the distance from $I$ to any side.
We use the distance from a point to a line: $d = \dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.
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 $4x+3y-5=0$.
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…
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.
Extension: the Euler line
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…
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 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…
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.
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…
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 ($HG = 2GO$).
Find the coordinates of vertices $B$ and $C$ of a triangle knowing that vertex $A$ is $(-3, 1)$, its circumcenter is the point $O(5, 2)$, and its centroid is $G(4, 2)$.
Find the coordinates of vertices $B$ and $C$ of a triangle knowing that vertex $A$ is $(-3, 1)$, its circumcenter is the point $O(5, 2)$, and its centroid is $G(4, 2)$.