Can Figure F be included in such a right-angle triangle T that S(T) = 2?S(F)?

1
Can Figure F be included in such a right-angle
triangle T that S(T) 2?S(F)?

• Author Andrejs Vihrovs, Riga, 2006

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Research aims

• To prove the hypothesis
• Each plain figure F can be included in such
• a right-angle triangle T that
• S(T) 2?S(F)
• for as many convex shapes as possible
• To calculate the minimum of the ratio S(T)S(F)
  for some basic shapes circle, regular triangle,
  regular hexagon.

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Background

• Some similar problems have been formulated and
  solved in the book
• ????????? ?. ?., ?????? ?. ?., ????? ?. ?,
  ?????????????? ?????? ? ?????? ?? ?????????????
  ?????????, ?., ?????, 1970, 384 c.
• Prove that any plain convex figure having area 1
  can be included into a parallelogram having area
  2 as well as into a triangle having area 2.
• Subsequently constraints on a triangle were
  added, namely it must be a right-angle triangle.
• Mцgling Werner, Ьber Trapezen umbeschriebene
  rechtwinklige Dreiecke, Wiss. Beitr.
  M.-Luther-Univ., Halle-Wittenberg, 1989, Nr. 56,
  161-170.

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Triangle

• Let a ? b ? c. Then the following arrangement
  satisfies the required inequality

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Quadrangle

• The estimation S(T) 2?S(F) has been proved for
  such quadrangles squares, rhombs, parallelograms
  and trapeziums. The arrangement of parallelogram
  is shown below, but it is also useful for rhomb
  and square.

Assumption b ? a Optimal angle
The ratio equals to 2.
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Trapezium

• The arrangement for isosceles trapezium with a gt
  b is shown below. S(T)S(F) 2.

Optimal angle
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Trapezium

• Arrangements for all the other types of
  trapeziums

b ? d or a gt d gt b ? d?cos ?
a gt d gt b and b lt d?cos ?
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Trapezium
d ? a and 0,5a ? b
d ? a gt b gt 0,5a and d ? 2b?cos ?
d ? a gt b gt 0,5a and c 2b?cos ?
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Regular polygon
The main idea is, firstly, to find the minimal
right-angle triangle T including the circle.
After having found T one determines all those
regular polygons Fn fitting in this circle and
satisfying S(T) 2?S(Fn). More precisely, it
has been shown that
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Regular polygon

• The arrangaments for the regular n-gons with
• 5 n 9 are shown below.
• The value of the ratio S(T)S(Fn) is given below
  the picture.

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Minimum of ratio

• Regular triangle and regular hexagon

Optimal angle
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Conclusions
It has been proved that the inequality S(T)
2?S(F) holds for all triangles, regular polygons
and trapeziums. The minimum of the ratio for the
circle, the regular triangle and the regular
hexagon are as follows

• The proof for any trapezium has been obtained
  independently and by techniques other than those
  of Prof. M. Werner.