Exercises: 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 $C(2, 6)$.

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

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

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

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

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

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 $4x+3y-5=0$.

In a 1st year of Bachillerato Mathematics exam, students are asked to find the general equation of the altitude from vertex $A$ in the triangle with vertices $A(1, 3)$, $B(-2, 1)$, and $C(4, -1)$.

Marcos submits the following solution:

1st Find the midpoint of side $BC$: $M = \left(\frac{-2+4}{2}, \frac{1+(-1)}{2}\right) = (1, 0)$.

2nd Calculate the slope of side $BC$: $m_{BC} = \frac{-1-1}{4-(-2)} = \frac{-2}{6} = -\frac{1}{3}$.

3rd Deduce the perpendicular slope: $m_\perp = 3$.

4th Write the line with slope $3$ passing through $M(1, 0)$: $y - 0 = 3(x - 1) \implies 3x - y - 3 = 0$.

Identify and explain in detail the conceptual error made by Marcos: which special line has he actually calculated instead of the altitude? Then, write out the correct procedure and find the general equation of the requested altitude.

In a triangle $\triangle ABC$, two of its vertices are known: $A(-3, 2)$ and $B(5, -4)$. It is also known that its centroid (center of gravity) is located at the point $G(1, 1)$.

1. a) Set up the necessary vector or analytical equations and determine, with reasoning, the coordinates of the third vertex $C$.
2. b) With the coordinates of the three vertices already calculated, classify the triangle according to its side lengths (equilateral, isosceles, or scalene), justifying your answer with the explicit calculation of their magnitudes.
3. c) Find the continuous equation of the line that contains the median from vertex $C$.

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 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 $C$.

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.

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 $C$.

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.

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.

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.

Consider the triangle with vertices $A(-4, 0)$, $B(4, 0)$, and $C(0, 6)$.

1. a) Estimate before calculating: Observing the coordinates and the symmetry of the triangle with respect to the y-axis, on which line must the centroid $G$, the orthocenter $H$, and the circumcenter $O$ necessarily lie? Furthermore, classify the angle at vertex $C$ (by calculating the sign of the dot product $\vec{CA} \cdot \vec{CB}$) to deduce whether the orthocenter and the circumcenter will lie inside or outside the triangle.
2. b) Calculate: Analytically obtain the coordinates of the circumcenter $O$ (using the intersection of perpendicular bisectors or the condition of equidistance to the vertices) and of the orthocenter $H$ (by intersecting two altitudes).
3. c) Check: Find the coordinates of the centroid $G$ using its direct formula and verify whether your estimates from part a) were correct. Numerically verify whether the Euler line relation $\vec{HG} = 2\,\vec{GO}$ holds.

The forest fire fighting service of Castilla-La Mancha (Plan INFOCAM) is analyzing the location of an advanced helitransported command post to coordinate three watchtowers located at points $A(1, 3)$, $B(8, 10)$, and $C(9, 3)$, whose coordinates are given in kilometers relative to the regional center. The technical committee is considering two criteria:

• Criterion 1 (Equidistance): Locate the post at the circumcenter $O$, so that the straight-line distance to any of the three towers is exactly the same.

• Criterion 2 (Center of mass): Locate the post at the centroid $G$, so that the average routine flight distance of the helicopter is minimized.

1. a) Model both criteria analytically by calculating the exact coordinates of the base according to Criterion 1 and according to Criterion 2.
2. b) If the reconnaissance helicopter has a guaranteed operational range of $6\text{ km}$ without refueling from its command post, determine with justification whether both locations allow all three towers to be reached within this safety limit.

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 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 $C$.

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 ($HG = 2GO$).

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.

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 coordinates of vertices $B$ and $C$.

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 $G(4, 2)$.

To calculate the coordinates of the circumcenter $O$ of the triangle with vertices $A(1, 5)$, $B(-1, 1)$, and $C(5, -2)$, three students propose different strategies:

• Strategy A: Obtain the equations of two perpendicular bisectors (by calculating midpoints and normal vectors) and solve the $2 \times 2$ linear system.

• Strategy B: Require the metric equality of distances $d(O, A)^2 = d(O, B)^2 = d(O, C)^2$ for a generic point $O(x, y)$, expand the standard algebraic identities, and solve the resulting linear system.

• Strategy C: First check the particular geometric properties of the triangle (calculating the dot products between its side vectors) in case it is a right triangle, in which case a theoretical property allows finding the circumcenter directly.

1. a) Critically compare strategies A and B: how many algebraic steps does each require, and in which one is it easier to make sign or calculation errors?
2. b) Apply Strategy C: calculate the vectors $\vec{BA}$ and $\vec{BC}$, compute their dot product $\vec{BA} \cdot \vec{BC}$, and interpret the obtained result geometrically.
3. c) Relying on the conclusion of part b), obtain the coordinates of the circumcenter $O$ with minimal calculation, and justify why this strategy is vastly superior to the other two for this specific triangle.

In a non-equilateral triangle $\triangle ABC$, the coordinates of vertices $B$ and $C$ have been lost from the working plane, but the following data are known with certainty:

• The circumcenter of the triangle is $O(1, -2)$.

• The orthocenter of the triangle is $H(7, 4)$.

• One of the vertices is $A(-1, 2)$.

1. a) Deduce, with reasoning, the coordinates of the centroid $G$ using the vector property of the Euler line ($\vec{HG} = 2\,\vec{GO}$), proving that it is not necessary to know vertices $B$ and $C$ beforehand.
2. b) Knowing that the centroid divides the median from $A$ in a $2:1$ ratio from that vertex (that is, $\vec{AG} = 2\,\vec{GM_A}$, where $M_A$ is the midpoint of side $BC$), find the coordinates of $M_A$.
3. c) Justify why the line passing through the circumcenter $O$ and the midpoint $M_A$ is necessarily the perpendicular bisector of segment $BC$. Write its general equation and deduce from it the slope of the line containing side $BC$.
4. d) Explain with reasoning how the coordinates of the two missing vertices, $B$ and $C$, could be determined precisely from the intersection of the line containing side $BC$ with the circumscribed circle of center $O$ and radius $R = d(O, A)$ (it is not necessary to solve the resulting quadratic equation; it is sufficient to formulate the procedure and justify why it yields exactly two solutions).

Topic challenge. Where would you locate an emergency heliport to provide optimal service to three municipalities in your province?

1. Data collection: Access the IGN viewer or Google Maps and choose three localities that form a triangle. Set a Cartesian coordinate origin (in km) to obtain the coordinates of the three vertices: $A$, $B$, and $C$.

2. Construction and notable lines: Transfer the points to GeoGebra. Analytically find the midpoints of the sides and the symmetric and standard forms of the equations of two perpendicular bisectors. Analyze their relative position and calculate their intersection point to find the circumcenter (equidistant evacuation center).

3. Metrics and angles: Determine the Euclidean distance from the heliport to each municipality and the distance to the line connecting $A$ and $B$. Also find the angle at vertex $A$ in both sexagesimal degrees and radians ($rad$) using the direction vectors of the sides.

4. Reasoned proposal: Prepare a one-page report with the GeoGebra plot justifying whether it is more convenient to build the base at the circumcenter or at the centroid.

Assessment criteria:

• Rigour in the analytical calculation of equations, distances, and angle conversions.

• Correctness in finding midpoints, notable points, and relative positions.

• Accurate use of GeoGebra and consistency in the chosen territorial justification.