Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 37

Exercises

Exercise

Let ${}E,F$ be affine spaces, and let

$\varphi \colon E\longrightarrow F$

denote an affine-linear mapping. Show that the center of gravity of the points ${}P_{1},\ldots ,P_{n}$ is transformed under ${}\varphi$ to the center of gravity of the image points ${}\varphi (P_{1}),\ldots ,\varphi (P_{n})$.

Exercise

Show that a triangle is equilateral if and only if its center of gravity coincides with the circumcenter.

Exercise

Show that there does not exist a nondegenerate equilateral triangle in ${}\mathbb {R} ^{2}$ such that all its vertices have rational coordinates.

Exercise

Show that the triangle of the midpoints of a triangle is similar to the original triangle.

Exercise

Determine the medians of the triangle with the vertices ${}(-1,3),(0,-5),(2,1)$ in ${}\mathbb {R} ^{2}$. Determine its center of gravity with several methods.

Exercise

Determine, for the triangle given by the standard vectors ${}e_{1},e_{2},e_{3}$ in ${}\mathbb {R} ^{3}$, the medians, and the center of gravity.

Exercise

Let ${}A,B$ be different points in a Euclidean plane. Show that the perpendicular bisector of ${}A$ and ${}B$ consists of all those points that have the same distance to ${}A$ and to ${}B$.

Exercise

Let ${}D$ be a nondegenerate triangle. Show that any two perpendicular bisectors are linearly independent.

Exercise

Determine, for the triangle with the vertices ${}(-2,2),(0,4),(5,0)$ in ${}\mathbb {R} ^{2}$, the perpendicular bisectors, the circumcenter, and the radius of the circumcircle.

Exercise

Let a nondegenerate triangle in ${}\mathbb {R} ^{2}$ be given, where the vertices have the coordinates

${\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}},\,{\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}},\,{\begin{pmatrix}x_{3}\\y_{3}\end{pmatrix}}.$

Let

${}d=2(x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}))\,.$

Show that the circumcenter of the triangle has the coordinates

${}x={\frac {(x_{1}^{2}+y_{1}^{2})(y_{2}-y_{3})+(x_{2}^{2}+y_{2}^{2})(y_{3}-y_{1})+(x_{3}^{2}+y_{3}^{2})(y_{1}-y_{2})}{d}}\,,$

and

${}y={\frac {(x_{1}^{2}+y_{1}^{2})(x_{3}-x_{2})+(x_{2}^{2}+y_{2}^{2})(x_{1}-x_{3})+(x_{3}^{2}+y_{3}^{2})(x_{2}-x_{1})}{d}}\,.$

Exercise

Show that the circumcenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is mapped under a translation and under an angle-preserving mapping to the circumcenter of the image triangle.

Exercise

Show that the circumcenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is not necessarily mapped to the circumcenter of the image triangle under a bijective affine-linear mapping.

Exercise

Let ${}v,w$ be linearly independent vectors in ${}\mathbb {R} ^{2}$. Show that the angle bisector of ${}v$ and ${}w$ defines with ${}v$ and with ${}w$ the same angle.

Exercise

Show that the incenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is mapped under a translation and under an angle-preserving mapping to the incenter of the image triangle.

Exercise

Show that the incenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is not necessarily mapped to the incenter of the image triangle under a bijective affine-linear mapping.

Exercise

Sketch a triangle where two feet of altitudes lie outside of the edges.

Exercise

Show that in a nondegenerate triangle at least one foot of the altitude lies between two vertices.

Exercise

Let ${}D$ be an isosceles triangle. Denote the vertex where the sides of the same length meet by ${}A$ and the opposite side by ${}a$. Show that the median through ${}A$, the angle bisector through ${}A$, the altitude through ${}A$, and the perpendicular bisector of ${}a$ coincide.

In the following exercises, we work with a naive concept of area. The area of a rectangle is the product of its side lengths, and the area fulfills the additive property (for disjoint regions) and the invariance under translations.

Exercise

Show that for a parallelogram, the area equals the length of the base times the height.

Exercise

Show that the area of a triangle equals one-half the product of height and base length.

Exercise

Show that the orthocenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is mapped under a translation and under an angle-preserving mapping to the orthocenter of the image triangle.

Exercise

Show that the orthocenter of a nondegenerate triangle ${}(A,B,C)$ in a Euclidean plane is not necessarily mapped to the orthocenter of the image triangle under a bijective affine-linear mapping.

Exercise

Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities

${}{\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}\,$

hold, where ${}a,b,c$ are the side lengths of the edges and ${}\alpha ,\beta ,\gamma$ are respectively the opposite angles.

Exercise

Let a triangle be given by the points ${}A,B,C$, denote its side lengths by ${}a,b,c$, and the angles by ${}\alpha ,\beta ,\gamma$. Let ${}F$ be the area of the triangle. Show that

${}{\frac {a}{\sin \alpha }}={\frac {abc}{2F}}\,$

holds.

In the following exercise, we prove a version of the Heron's formula.

Exercise

Let ${}a,b,c$ be the side lengths of a triangle. Show that the area of the triangle equals

${}F={\frac {\sqrt {2a^{2}c^{2}+2c^{2}b^{2}+2a^{2}b^{2}-a^{4}-b^{4}-c^{4}}}{4}}\,.$

Exercise

Let ${}D$ be a nondegenerate triangle in the plane, given by the three vertices ${}A,B,C$. Show that the altitudes, the perpendicular bisectors, the angle bisectors, and the medians can be constructed with ruler and compass.

Exercise

Suppose that a triangle has the base side ${}[0,s]$, and the height ${}h$ (${}s,h>0$). For which foot of the altitude ${}x$ has the triangle a minimal perimeter; how long is it?

Exercise

We consider the set of all (also degenerate, ordered) triangles ${}\triangle =(A,B,C)$ in ${}\mathbb {R} ^{2}$, using its coordinates ${}A=(A_{1},A_{2}),B=(B_{1},B_{2}),C=(C_{1},C_{2})$, as the vector space ${}\mathbb {R} ^{6}$. In particular, we can add triangles, and we can multiply a triangle with a scalar ${}s\in \mathbb {R}$.

a) Show that the triangles ${}\triangle$ and ${}s\triangle$, where ${}\triangle$ is nondegenerate and ${}s\neq 0$, are similar to each other.

b) Let ${}S$ be the center of gravity of the triangle ${}(A,B,C)$. Show that the triangles

$(A,B,C),\,(B,C,A),\,(C,A,B){\text{ and }}(S,S,S)$

are linearly dependent.

c) Determine whether the following sets of triangles form a linear subspace, or not. If yes, determine its dimension.

1. The set of all nondegenerate triangles.
2. The set of all triangles with ${}0$ as first vertex.
3. The set of all triangles with center of gravity ${}0$.
4. The set of all equilateral triangles.
5. The set of all triangles with the property that their circumcircle is the unit circle.
6. The set of all triangles contracted to a point.
7. The set of all right triangles.
8. The set of all right triangle where the right angle is at the first vertex in ${}0$, and where the second vertex lies on the ${}x$-axis.
9. The set of all triangles where the orthocenter is ${}0$.

Hand-in-exercises

Exercise (4 marks)

Determine, for the triangle ${}(0,0),(3,0),(0,4)$, the center of gravity, the circumcenter, the incenter, and the orthocenter.

Exercise (3 marks)

Determine the Euler line for the triangle ${}(2,3),(1,8),(6,-5)$.

Exercise (4 marks)

Determine, for the triangle given by ${}(4,-3),(7,2),(-1,5)$, the center and the radius of its Nine-point circle.

Exercise (4 marks)

Determine, for the triangle given by the vectors

${\begin{pmatrix}2\\5\\6\end{pmatrix}},\,{\begin{pmatrix}1\\4\\9\end{pmatrix}},\,{\begin{pmatrix}3\\3\\5\end{pmatrix}}$

in ${}\mathbb {R} ^{3}$, the altitude through ${}{\begin{pmatrix}2\\5\\6\end{pmatrix}}$, and the area of the triangle.

Exercise (2 marks)

We consider the vector space of all triangles in ${}\mathbb {R} ^{2}$ as in Exercise 37.28 . Is the mapping that assigns to a triangle its perimeter, a linear form?

In the following exercise, we refer to the concept of convergence of sequences in ${}\mathbb {R} ^{2}$. Convergence holds if and only if the two component sequences in ${}\mathbb {R}$ converge.

Exercise (6 marks)

For a triangle ${}\triangle =(A,B,C)$, its triangle of midpoints is given by the vertices ${}{\frac {1}{2}}(A+B),{\frac {1}{2}}(A+C),{\frac {1}{2}}(B+C)$. This construction yields a recursively defined sequence of triangles ${}\triangle _{n}$, where ${}\triangle _{1}=\triangle$, and ${}\triangle _{n+1}$ is the triangle of midpoints of ${}\triangle _{n}$. Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a sequence in ${}\mathbb {R} ^{2}$ with ${}x_{n}\in \triangle _{n}$ for all ${}n\in \mathbb {N} _{+}$. Show that this sequence converges, and determine its limit.

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