You cut a triangle out of cardstock and want to balance it on the tip of a pencil. At which point do you rest it? That point, and three others just as special, can be calculated with the lines you already know.
Centroid
Why the average? $G$ lies on each median at two thirds of the way from the vertex. That is why its position is the average of the three vertex positions, just like the balance point of three equal weights.
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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 $C(-1, 5)$.
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 $C$.
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.
Go back to the beginning: the circumcenter is the point used to place something (a sprinkler, an antenna) at the same distance from three locations.
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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 $B(-4, 1)$.
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.
To equate distances you need the distance from a point to a line.
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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.
Thanks to this property, knowing two of the three points is enough to find the third: $G = \dfrac{H + 2O}{3}$ in coordinates.
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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 ($HG = 2GO$).
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.
Sometimes you know the notable point and you are missing a vertex. The idea is to use the property of the point in reverse.
Centroid: from $G = \dfrac{A + B + C}{3}$ you get $C = 3G - A - B$, coordinate by coordinate.
Orthocenter: $C$ is on the line through $H$ perpendicular to $AB$, and also on the line through $B$ perpendicular to $AH$.
Circumcenter and centroid: $B + C = 3G - A$ gives the midpoint of $BC$; $BC$ is perpendicular to $OM$; and $B$ and $C$ are at distance $OA$ from $O$.
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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.
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 $C$.
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 $G(4, 2)$.
If you are given one vertex and the equations of two notable lines, place each sought vertex on its line and use what you know about the line.
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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 $C$.
If a vertex depends on a parameter $t$, so does the notable point. Its path is a locus: you find it by writing $x$ and $y$ as functions of $t$ and eliminating $t$.
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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.
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.