Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. Expected questions involve calculating lengths, angles, areas, and volumes of various shapes, understanding geometric theorems, and solving problems related to spatial reasoning.

Solid Geometry / Geometry in Space Trigonometry Spherical Geometry Plane Geometry Vectors

  • Question

    Consider an arbitrary triangle. Draw tangents to the inscribed circle parallel to the sides of the triangle. These tangents cut off three smaller triangles from the original triangle. Prove that the sum of the radii of the inscribed circles of these smaller triangles is equal to the radius of the inscribed circle of the original triangle.

  • Question

    Consider an arbitrary hexagon and denote the midpoints of its sides by `M_1,M_2,M_3,M_4,M_5,M_6`. Prove that the segments `M_1M_2`, `M_3M_4` and `M_5M_6` can form a triangle, even without rotating these segments.

    Topics:
    Geometry -> Vectors

  • A Walk in the Plane

    Given a Cartesian coordinate system x-y in the plane. You need to get from the point (1,0) to the point (2006,2005), where in each step you are allowed to move one unit up (in the positive direction of y) or one unit to the right (in the positive direction of the x-axis).

    a. In how many different paths can the task be performed?

    b. In how many different paths can the task be performed if it is forbidden at any stage to pass through a point on the line x=y?

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  • THE MILKMAID PUZZLE

    Here is a little pastoral puzzle that the reader may, at first sight, be led into supposing is very profound, involving deep calculations. He may even say that it is quite impossible to give any answer unless we are told something definite as to the distances. And yet it is really quite "childlike and bland."

     

    In the corner of a field is seen a milkmaid milking a cow, and on the other side of the field is the dairy where the extract has to be deposited. But it has been noticed that the young woman always goes down to the river with her pail before returning to the dairy. Here the suspicious reader will perhaps ask why she pays these visits to the river. I can only reply that it is no business of ours. The alleged milk is entirely for local consumption.

    "Where are you going to, my pretty maid?"
    "Down to the river, sir," she said.
    "I'll not choose your dairy, my pretty maid."
    "Nobody axed you, sir," she said.


    If one had any curiosity in the matter, such an independent spirit would entirely disarm one. So we will pass from the point of commercial morality to the subject of the puzzle.

    Draw a line from the milking-stool down to the river and thence to the door of the dairy, which shall indicate the shortest possible route for the milkmaid. That is all. It is quite easy to indicate the exact spot on the bank of the river to which she should direct her steps if she wants as short a walk as possible. Can you find that spot?

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  • THE FLY ON THE OCTAHEDRON

    "Look here," said the professor to his colleague, "I have been watching that fly on the octahedron, and it confines its walks entirely to the edges. What can be its reason for avoiding the sides?"

    "Perhaps it is trying to solve some route problem," suggested the other. "Supposing it to start from the top point, how many different routes are there by which it may walk over all the edges, without ever going twice along the same edge in any route?"

     

    The problem was a harder one than they expected, and after working at it during leisure moments for several days their results did not agree—in fact, they were both wrong. If the reader is surprised at their failure, let him attempt the little puzzle himself. I will just explain that the octahedron is one of the five regular, or Platonic, bodies, and is contained under eight equal and equilateral triangles. If you cut out the two pieces of cardboard of the shape shown in the margin of the illustration, cut half through along the dotted lines and then bend them and put them together, you will have a perfect octahedron. In any route over all the edges it will be found that the fly must end at the point of departure at the top.

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  • THE ICOSAHEDRON PUZZLE

    The icosahedron is another of the five regular, or Platonic, bodies having all their sides, angles, and planes similar and equal. It is bounded by twenty similar equilateral triangles. If you cut out a piece of cardboard of the form shown in the smaller diagram, and cut half through along the dotted lines, it will fold up and form a perfect icosahedron.

    Now, a Platonic body does not mean a heavenly body; but it will suit the purpose of our puzzle if we suppose there to be a habitable planet of this shape. We will also suppose that, owing to a superfluity of water, the only dry land is along the edges, and that the inhabitants have no knowledge of navigation. If every one of those edges is `10,000` miles long and a solitary traveller is placed at the North Pole (the highest point shown), how far will he have to travel before he will have visited every habitable part of the planet—that is, have traversed every one of the edges?

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