Can Figure F be included in such a right-angle triangle T that

S(T) ≤ 2S(F)?

Author: Andrejs Vihrovs, Riga, 2006

Research aims To prove the hypothesis:

Each plain figure F can be included in such

a right-angle triangle T that

S(T) ≤ 2S(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.

BackgroundSome similar problems have been formulated and

solved in the book Шклярский Д. О., Ченцов Н. Н., Яглом И. М, Геометрические

оценки и задачи из комбинаторной геометрии, М., Наука, 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.

Mögling Werner, Über Trapezen umbeschriebene rechtwinklige Dreiecke, Wiss. Beitr. M.-Luther-Univ., Halle-Wittenberg, 1989, Nr. 56, 161-170.

TriangleLet a b c. Then the following arrangement

satisfies the required inequality

S(T)1 2

S(F)

a

b

c

Assumption:

b a

Optimal angle

QuadrangleThe estimation S(T) ≤ 2S(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.

bsinaarctg

The ratio equals to 2.

a

b

TrapeziumThe arrangement for isosceles trapezium with a > b is

shown below. S(T):S(F) = 2.

Optimal angle

coscbsincarctg

a

b

c

TrapeziumArrangements for all the other types of trapeziums:

b d or a > d > b dcos

d

b

a

ad

b

a > d > b and b < dcos

Trapezium

d a and 0,5a b

d a > b > 0,5a and d 2bcos

d a > b > 0,5a and c ≤ 2bcos

b

d

a a

d

ba

bd

c

Regular polygonThe 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) ≤ 2S(Fn).More precisely, it has been shown that

S(T) 3 2 21,855...

S(Circle)

0

n

S(T)2 10

S(F )n n

Calculation of n0:

223π2 n

sinn

Regular polygonThe 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.

5

1.8005... 6

1.9175...8

1.7588...

1.9437...

7

1.9068...

9

Minimum of ratioRegular triangle and regular hexagon

045

6

S(T) 12 13 31,917...

S(F ) 18

3

S(T)3

S(F )

Optimal angle

Conclusions

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

18

31312;3;

π

223

It has been proved that the inequalityS(T) ≤ 2S(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: