Cut a triangle out of cardboard and try to balance it on the tip of a pencil. Exactly one spot works. Now picture a town planner who wants a fountain that is the same distance from all three entrances of a park. Is that spot special too? Every triangle hides four special points like these, and you can find all of them with the line equations you already know.
Medians and the centroid
Tools you will use all the time
Some problems ask for the explicit equation, which is slope-intercept form $y = mx + b$. Others ask for the general equation, which is $Ax + By + C = 0$. They describe the same line, so you can convert between them.
Medians
Why is it an average? The centroid is the "center of mass" of the three corners, and a center of mass is just the average position.
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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 the 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 orthocenter 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 orthocenter of this triangle.
This is the park-fountain point from the opening. A fountain at $O$ is the same distance from all three entrances.
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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 $B(-4, 1)$.
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 $B(-4, 1)$.
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…
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.
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 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 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.
The second form comes from $G - H = 2(O - G)$. If you know two of the three points, you get the third from it.
Let's use the triangle $A(0, 0)$, $B(6, 0)$, $C(2, 4)$ from the last two sections. We already found $H = (2, 2)$ and $O = (3, 1)$.
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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 orthocenter. 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 orthocenter. Next, obtain the general equation of the Euler line that passes through both points.
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…
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 ($HG = 2GO$).
Sometimes the problem runs backward. You know one or two vertices and a special point, and you must find the third vertex. The idea is always the same. Write down what the special point tells you, then solve.
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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 the 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 the vertex $C$ and the Cartesian equation of the median starting from this vertex.
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 $C$.
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 $C$.
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…
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.
Now the data are equations of lines instead of points. Three ideas do all the work.
Practice★★★Not started
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…
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 $C$.
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…
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 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 $C$.
To prove something for every triangle of a family, use letters for the coordinates. Place the triangle on the axes so the numbers stay simple, and keep the letters positive when you can.
A locus is the path traced by a point that moves according to some rule. When a vertex depends on a parameter $t$, write the point's coordinates in terms of $t$, then eliminate $t$.
Practice★★★Not started
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…
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…
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.
Back to the start: the cardboard balances at the centroid, which is the average of the three vertices. The fountain that is the same distance from all three entrances is the circumcenter, where the perpendicular bisectors of the sides meet.
Summary
Let's practice! 8 medium exercises in this section