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the same! Pitot's theorem must hold in both so the quadrilateral is tangential (has an incircle). The other condition is a little different. For a tangential trapezoid it is that adjacent angles are supplementary, whereas in a bicentric quadrilateral it is that opposite angles are supplementary. That
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Further, you are correct that a tangential trapezoid is not always bicentric. If a convex quadrilateral shall be a bicentric trapezoid, then it shall satisfy three conditions: Pitot's, adjacent angles supplementary and opposite angles supplementary. Then it is a tangential isosceles trapezoid, as in
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Oh, yes. I didn't read well and haven't seen words 'opposite' and 'adjacent' in both characterizations; and also both second formuale for angles, which of course differ. For sure - it makes sense then. Perhaps what should also can be noted is that in tangential trapezoid (or in any other trapezoid)
852:. Hence the problem you address. It would of course be prefarable to be able to cite a reference giving these formulas, but this is a special case that does not seem to appear often in the litterature. Please check my algebra so it is correct this time, and lets be on the look out for a reference.
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But not every tangential trapezoid is (also) bicentric (only isosceles (tangential) ones). So, what is wrong here? A quadrilateral with these two properties is not automatically (also) bicentric - am I right? I guess something is missing in 1st characterization for bicentric quadrilateral.
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You are correct! There was in fact two errors, both the letters and the factor 2 in the denominator. I wrote the first draft for this article and the fault was mine. I didn't find a reference for these formulas, so I used the formula for the area of a trapezoid
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A convex quadrilateral is a tangential trapezoid if and only if opposite sides satisfy Pitot's theorem (so it is tangential) and it has two adjacent angles that are supplementary (then this is also true for the other two angles) (so it is a trapezoid). Hence
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does not hold - this can be perhaps well seen in image for right tangential trapezoid. Also I think that now citations for area and inradius formulae are not needed anymore, as they can be derived from area formula for trapezoid.
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Both of these involve the square root of a negative number, so they can't be right. Unfortunately no citation is given, so I can't look up the correct version. Does anyone know it?
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is the difference between that opposite sides are parallel (trapezoid) or that the quadrilateral has a circumcircle (bicentric).
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and simplified it using Pitot's theorem. The problem is that the sides in a trapezoid are in consecutive order:
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as in most (all) other quadrilaterals. Thus Pitot's theorem in a tangential trapezoid states
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At bicentric quadrilateral a similar one (if not the same) characterization is given:
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holds), not perhaps on bases, so relation for other pair of adjacent angles
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There is nothing wrong. As you said yourself, they are similar but
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is bicentric if and only if opposite sides satisfy Pitot's theorem
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Here this characterization of tangential trapezoid is given:
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