  # becko

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Lepton
1. This construction uses the straightedge (to draw the line between two points at step 3 and to find the point of intersection of this line with AB at step 4). I need a construction that uses only the Euclidean compass, that doesn't transfer distances.
2. I saw the website. There's a construction for the middlepoint of the arc (http://www.cut-the-knot.org/do_you_know/compass7.shtml). But this construction uses the compass to transfer distances (when the circles with radius OE and centers at C and D are drawn; we cannot transfer the distance OE and use it as a radius to draw a circle around C or D). I've already convinced myself that by the use solely of a compass that transfers distances, you can accomplish all the constructions allowed by the use of straightedge and compass. My question is whether these constructions can be achieved by a compass that doesn't transfer distances (the classical Euclidean compass that brakes if lifted from the paper). I should make a small change in my previus description of the fundamental constructions allowed by the compass: To find the points of intersection of two circles. To draw a circle with given center passing through a given point. Now it is clear that this compass cannot be used to transfer distances. All I need is a valid construction of the arc bisector. If I have this construction, I can prove that all the constructions allowed by straightedge and compass can be done using the Euclidean compass alone.
3. In geometrical constructions using compass alone, we should restrict ourselves to two fundamental constructions: To find the points of intersection of two circles. To draw a circle with given center and radius. Can anyone tell me how to find the middlepoint of a given arc using only a compass?
4. I haven't had any advance. I'd appreciate any ideas. Maybe if we assume that one of the numbers is a perfect square, so that 4(a^2) + r(b) = k^2, where k is a positive integer, then as a consequence, 4(b^2) + r(a) shouldn't be a perfect square. Or if we assume that both numbers are perfect squares, we would surely arrive at some contradiction. I have tried to do both ways, but I have got nowhere. All I have are some crumpled papers with nothing valuable in them. Maybe someone can show me how to prove this?
5. I cannot find a way to prove this: A function r(n), where n is a positive integer, gives the number n with its digits inverted. Prove that for positives integers a and b, the two numbers 4(a^2) + r(b) and 4(b^2) + r(a) cannot be simultaneously perfect squares. Can anyone help me?
6. I'm looking for information concerning the physiology of the hand adressed to piano performance. References to books, articles, web sites, etc, would be greatly appreciated.
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