Middle East Technical University Geometry II Math 374 Spring 2008 Instructor: S. Finashin

LECTURE 1. ISOMETRY AND

SIMILARITY TRANSFORMATIONS

1. Klein’s definition of Geometry. Symmetry in nature: in time (periodicity), music, dancing (palindromes, tessellations, patterns, tiling, wallpapers, kaleidoscopes, carpets). Wallpaper groups. Linear and affine isometries. Classification of isometries, composition. Congruence of figures. Orientation. Coordinate change.

1.1. Exercise. Find the generators of the symmetry groups of some figure (tetrahedron, cube, etc.), some sign (asterisk, cross, suits of cards) and some pattern (Escer, etc.).

1.2. Problem. Prove that an isometry should send lines into lines.

1.3. Problem. Prove that each isometry is determined by the images of any three points which are not collinear.

1.4. Problem. Prove that each isometry can be presented as a composition of a linear isometry and a translation.

2. Translations. Coordinate change: (x, y) ↦ (x + a, y + b). Translational symmetry.

2.1. Exercise. Sketch the new coordinate axes, x′, y′ after the translation (x, y) ↦ (x − 1, y + 2). Find the equations of the new axes.

2.2. Problem. (a) Two villages are separated by a river of a constant width. Find the place to construct a bridge which minimizes the total length of the way between the villages.

(b) Solve a similar problem if the villages are separated by several rivers.

2.3. Problem. (a) Given two circles, S_{1}, S_{2}, and a line ℓ, draw a line ℓ′ parallel to ℓ, so that the distance between the intersection points of ℓ′ with S_{1} and S_{2} has a given value a.

(b) How many different answers (that is such lines ℓ′) to this problems may exist ?

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LECTURE 1. ISOMETRY AND SIMILARITY TRANSFORMATIONS

2.4. Problem. Assume that M, N are the midpoints of the sides AD, BC of a quadrilateral ABCD. Knowing that |MN| = \frac{1}{2}(|AD| + |BC|), prove that ABCD is a trapezoid.

3. Rotations. Coordinate change: (r, θ) ↦ (r, θ + ϕ), (x, y) ↦ (x cos ϕ − y sin ϕ, x sin ϕ+y cos ϕ). Special case: central symmetry. Rotational symmetry of a figure, the center of symmetry. Composition of two rotations; special case: composition of two symmetries. Composition of a rotation and a translation.

3.1. Exercise. Find the coordinate change formulas for the rotation (1) by π/2 around P = (1, 0);

(2) by −π/2 around P = (1, 1);

(3) by π/4 around P = (1, −1).

3.2 Exercise. Show that (x, y) ↦ (y − 2, 3 − x) describes an isometry, F : ℝ^{2} → ℝ^{2}. Present F as a composition

(1) F = T ∘ R;

(2) F = R ∘ T , of a translation T and a rotation R (in each case determine the coordinate change formulas for R and T).

3.3. Exercise. Describe the type and find the coordinate change formulas for F = R_{1} ∘ R_{2}, where R_{1} is the rotation around (1, 0) by angle π/6 and R_{2} is the rotation around (3, 0) by angle −π/6.

3.4. Exercise. (a) Prove that a composition F = C_{1} ∘ ⋯ ∘ C_{n} of n central symmetries, C_{i}, is a translation if n is even, and a central symmetry if n is odd.

(b) Determine the center of the symmetry F, if n = 3 and C_{1}, C_{2}, C_{3} have centers of symmetry at (0, 1), (1, 0), and (1, 1) respectively.

(c) Determine the translation vector, if n = 4 and C_{1}, C_{2}, C_{3} are as above and C_{4} has the center at (2, 3).

3.5. Problem. (a) Given a point A, line ℓ and a circle S, find a line ℓ′ passing through A, so that A is the midpoint of a line segment BC, where B, C are the intersection points of ℓ′ with ℓ and S respectively.

(b) Does a solution always exist ? If it exists, how many different answers are possible ?

3.6. Problem. The region between two parallel lines has obviously infinitely many centers of symmetry. Is it possible for a figure to have more than one, but finitely many centers of symmetry (for example, two and only two centers) ?

3.7. Problem. (a) Show that the midpoints of the sides of a quadrilateral form a parallelogram.

(b) Knowing the midpoints, M_{1}, …, M_{5}, of pentagon’s sides, determine its vertices.

(c) Show that (b) admits one and only one solution for any given points M_{i}.

3.8. Problem. Construct an equilateral triangle whose vertices lie on three given parallel lines.

3.9. Problem. Prove that

cos(ϕ) + cos(ϕ + α) + cos(ϕ + 2α) + ⋯ + cos(ϕ + (n − 1)α) = 0

where α = \frac{2π}{n}, n ∈ ℕ, ϕ ∈ ℝ.

3.10. Problem. For an arbitrary triangle ABC, we construct equilateral triangles on its sides (looking outward). Show that their centers, O_{1}, O_{2}, O_{3} form also an equilateral triangle.

3.11. Problem. For an arbitrary parallelogram ABCD, we construct squares on its sides (looking outward). Show that their centers, O_{1}, O_{2}, O_{3}, O_{4} form also a square.

4. Bilateral (reflection) symmetry. Isosceles triangles and trapezoids, kite (deltoid), dart, rhombus. Composition with a translation and with a rotation. Glide reflection. Chirality. Composition of several bilateral symmetries.

4.1. Exercise. (a) Give the coordinate change formulas for compositions F_{1} = R_{1} ∘ R_{2} and F_{2} = R_{2} ∘ R_{1}, of the reflection R_{1}, against line x = y and R_{2}, against line y = 1.

(b) Determine the geometric meaning of isometries R_{1} and R_{2}.

4.2. Problem. Given points A, B from the same side of a line ℓ, find a point P ∈ ℓ such that |P A| + |P B| takes its minimal value.

4.3. Problem. Find the direction to hit a billiard ball so that it will hit another ball after reflecting against the four billiard borders.

4.4. Problem. Show that any isometry F can be presented as a composition of

(1) two reflections if F is orientation preserving;

(2) one or three reflections, if F is orientation reversing.

5. Similarity. Similarity transformation (dilation, homothety, homothecy). Properties.

5.1. Problem. Prove that every similarity transformation other that a translation has a fixed point.

5.2. Exercise. Prove that a composition F = T ∘ H of the homothety H(x, y) = (2x, 2y) with a translation T (x, y) = (x + 2, y + 4) is a dilation. What is its coefficient and the center ?

5.2. Problem. (a) Explain the nine-point circle.

(b) Explain the Euler line: which the centroid, the circumcenter, the orthocenter and the center of the nine-point circle lie on the same line.