Geometry, Plane Geometry, Symmetry

Symmetry in geometry refers to a shape or object remaining unchanged under certain transformations like reflection (line symmetry) or rotation (rotational symmetry). Questions involve identifying types of symmetry, lines of symmetry, and centers/orders of rotation for various figures.

• Where is the point?

  In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 3

• Question

  Inside a square ABCD with side length 1, a point E is marked, and outside the square, a point F is marked, such that triangles ABE and DAF are equilateral. Calculate the area of the pentagon CBEFD.
  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 6

• THE CHRISTMAS PUDDING

  "Speaking of Christmas puddings," said the host, as he glanced at the imposing delicacy at the other end of the table. "I am reminded of the fact that a friend gave me a new puzzle the other day respecting one. Here it is," he added, diving into his breast pocket.

  "'Problem: To find the contents,' I suppose," said the Eton boy.

  "No; the proof of that is in the eating. I will read you the conditions."

   

  "'Cut the pudding into two parts, each of exactly the same size and shape, without touching any of the plums. The pudding is to be regarded as a flat disc, not as a sphere.'"

  "Why should you regard a Christmas pudding as a disc? And why should any reasonable person ever wish to make such an accurate division?" asked the cynic.

  "It is just a puzzle—a problem in dissection." All in turn had a look at the puzzle, but nobody succeeded in solving it. It is a little difficult unless you are acquainted with the principle involved in the making of such puddings, but easy enough when you know how it is done.

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 168

• BOARDS WITH AN ODD NUMBER OF SQUARES

  We will here consider the question of those boards that contain an odd number of squares. We will suppose that the central square is first cut out, so as to leave an even number of squares for division. Now, it is obvious that a square three by three can only be divided in one way, as shown in Fig. `1`. It will be seen that the pieces A and B are of the same size and shape, and that any other way of cutting would only produce the same shaped pieces, so remember that these variations are not counted as different ways. The puzzle I propose is to cut the board five by five (Fig. `2`) into two pieces of the same size and shape in as many different ways as possible. I have shown in the illustration one way of doing it. How many different ways are there altogether? A piece which when turned over resembles another piece is not considered to be of a different shape.
  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 290

• THE CIGAR PUZZLE

  I once propounded the following puzzle in a London club, and for a considerable period it absorbed the attention of the members. They could make nothing of it, and considered it quite impossible of solution. And yet, as I shall show, the answer is remarkably simple.

  Two men are seated at a square-topped table. One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another. Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner? The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than `2` feet square and the cigar not more than `4`½ inches long. With those restrictions you may take any dimensions you like. Of course we assume that all the cigars are exactly alike in every respect. Should the first player, or the second player, win?

  Topics:

  Combinatorics -> Game Theory Logic -> Reasoning / Logic Geometry -> Plane Geometry -> Symmetry
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 398