Geometry Puzzles from the 35-th Lomonosov Tournament

Problem 1. It is true that it is possible to put positive numbers at the vertices of a triangle so that the sum of two numbers at the ends of each side is equal to the length of the side?

This looks like a simple linear algebra question with three variables and three equations. But it has a pretty geometrical solution. What is it?

Problem 2. Prove that it is possible to assign a number to every edge of a tetrahedron so that the sum of the three numbers on the edges of every face is equal to the area of the face.

Again we have six sides and four faces. There should be many solutions. Can you find a geometric one?

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6 Comments

  1. Carl Feynman:

    The first problem is solved by an equilateral triangle with the same number at each corner, and the second is solved by a regular tetrahedron with the same number on each edge. This is too easy. Surely I have misunderstood the problem somehow.

  2. tanyakh:

    Carl, The task is to do this for any triangle.

  3. none:

    At a glance seems quite easy
    1. Let a, b, c be the sides, p=a+b+c. Set values to (p-a)/2, (p-b)/2, (p-c)/2. The sum of, say, the first two is exactly c.
    2. Same idea. Too lazy for a school grad formula, but can easily imagine this with integrals. For each edge PQ assing \int_{[P,Q]}dm, where m denotes the measure over the manifold consisting of that edge.

  4. Alan:

    For the first one, the numbers would correspond to the radii of 3 touching circles centered at each vertex. The points where they touch would be the points of tangency of the inscribed circle of the triangle. http://demonstrations.wolfram.com/PairwiseTangentCirclesCenteredAtTheVerticesOfATriangle/

  5. lvps1000vm:

    I composed a solution example at the Ancient Greek Geometry game:

    http://sciencevsmagic.net/geo/#0A1.1A0.N.0L1.1L5.0L2.2L5.5A2.0A7.14A2.2A14.26L27.N.2A32.5A49.0A32

    (press several times Z-undo and then X-redo to see the steps)

  6. Zet:

    For the 1. problem, construct the incircle. The positive number on each vertice is the distance between the vertice and one of the two points where the incircle touches one of the adjacent edges.

    For the 2. problem, construct the inscribed sphere. The number of each edge is the area of the triangle built by the edge and one of the two points where the sphere touches one of the adjacent faces.

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