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B Trying to represent a written geometrical description

  1. Sep 28, 2016 #1
    I'm reading an old book about Thales (Greek geometry), and I can't understand what the next part means, and how to represent it graphically, could you help me? thanks:

    It begin stating that if you divide an equilateral traingle with a perpendicular from a vertex on the opposite side, it'll be divided into two right-angled triangles, equal to each other. Hence the sum of the three angles of each triangle is two right angles. (I can understand this)...then: "If now we suppose that Thales was led to examine whether the property, which he had observed in two distinct kinds of right-angled triangles, held generally for all right-angled triangles, it seems to me that, by completing the rectangle and drawing the second diagonal, he could easily see that the diagonals are equal, that they bisect each other, and that the vertical angle of the right-angled triangle is equal to the sum of the base angles."

    (by the property, refers to rule that the sum of all angles of a triangle is igual to two right angles).

    So what would it be to "complete the rectangle"?
    And, what would it be the vertical angle of the "right-angled triangle"? and the "base angles"?

    Just in case, the book is from 1889, so the copyrights are outdated.
  2. jcsd
  3. Sep 28, 2016 #2


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    Let the right-angled triangle have vertices at points (0,0), (0,b) and (a,0). Then completing the rectangle means adding a point at (a,b). The first diagonal is the hypotenuse of the first triangle and the 'second diagonal' is the line segment from (0,0) to (a,b).

    The term 'the vertical angle' is obscure but I think by that he is referring to the angle opposite the hypotenuse (ie the right angle) and 'the two base angles' refers to the two other angles (non-right angles).
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