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The pancake theorem guarantees that there is a line that can bisect a rectangle and a triangle simultaneously. The Borsuk Ulam Theorem can be used to prove that theorem. Show that any line through the center of a rectangle bisects the area of triangle simultaneously (the position of the center triangle can be the same as position the center of a rectangle or not)?

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Rohit Singh
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Dea
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    When you say "construct", do you mean the construction with the ruler and compasses? –  Jun 22 '23 at 06:26
  • (You seem to have most of the ideas needed, in fact much more than that. Try for something simpler instead of exotic.) Hint: Show that any line through the center of a rectangle bisects the area. Likewise for the centroid of a triangle. $\quad$ Thus, we can show that such a line exists, and even construct it (via ruler and compass). $\quad$ In fact, the hints are iff statements, and so the line is unique (unless the centers coincide). – Calvin Lin Jun 22 '23 at 07:19
  • @CalvinLin Thank you. If the centers triangle coincide with rectangle, the line should be also the median of triangle. Should we map that points on rectangle and triangle on circle (trigonometric function) to prove there is an antipodal point? My prof gave me a clue about the squezee theorem. But i didnt find the relationship between it. – Dea Jun 22 '23 at 11:26
  • Your question is going too far. It should be "Imagine an equilateral triangle and a random point, somewhere on one of the sides of that triangle. Is there a way to construct a line, going through that point, cutting the surface of that triangle in half?". When you have a solution of that problem, you can extend it to a point next to your triangle and later you can decide that point to be the centre of a rectangle. – Dominique Jun 22 '23 at 11:31
  • @Dominique, for a single triangle there is an infinite line can bisect triangle's area. The line should be tangent of hyperbola which both side of triangle as asymtote. – Dea Jun 22 '23 at 11:41
  • @Dea: I was looking for a way to do this, using ruler and compass :-) – Dominique Jun 22 '23 at 11:47

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