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Title
Abstract
Figures
Integer-Related Triangles
Other Integer-Related Figures
Method of Infinite Descent
The Case N = 8
References
All Topics
Mathematics
Geometry

Geometry

Profile image of Ahmad Nurali HilmanAhmad Nurali Hilman

Abstract

challenges in geometry

Figures (73)
Fig. 1.1 Pythagoras’ theorem.
Fig. 1.1 Pythagoras’ theorem.
Table 1.1 Primitive Pythagorean triples with c < 100.
Table 1.1 Primitive Pythagorean triples with c < 100.
Fig. 1.4 (a) The acute-angled Heron triangle, and (b) the obtuse-angled Heron triangle, with shortest side lengths.
Fig. 1.4 (a) The acute-angled Heron triangle, and (b) the obtuse-angled Heron triangle, with shortest side lengths.
Table 1.3 Primitive Pythagorean quartets with d < 20.
Table 1.3 Primitive Pythagorean quartets with d < 20.
Fig. 1.8 The five equable triangles. Table 1.4 The five equable triangles.
Fig. 1.8 The five equable triangles. Table 1.4 The five equable triangles.
Fig. 2.4 A cyclic quadrilateral with integer sides, integer diagonals, and integer radius.
Fig. 2.4 A cyclic quadrilateral with integer sides, integer diagonals, and integer radius.
Fig. 2.5 An inscribable quadrilateral with integer sides and integer radius.
Fig. 2.5 An inscribable quadrilateral with integer sides and integer radius.
Fig. 2.6 A cyclic inscribable quadrilateral.
Fig. 2.6 A cyclic inscribable quadrilateral.
Fig. 2.7 A triangle with integer sides and integer medians.
Fig. 2.7 A triangle with integer sides and integer medians.
Table 2.1 Integer-sided scalene triangles with a = 2A, b = 2B, and c = 2C and intege medians /, m, and n.
Table 2.1 Integer-sided scalene triangles with a = 2A, b = 2B, and c = 2C and intege medians /, m, and n.
Table 2.1 continued
Table 2.1 continued
Fig. 2.9 A triangle with an internal bisector of integer length
Fig. 2.9 A triangle with an internal bisector of integer length
Fig. 2.10 A triangle with integer sides and rational angle bisectors.
Fig. 2.10 A triangle with integer sides and rational angle bisectors.
Fig. 4.4 A cubic with two double points.
Fig. 4.4 A cubic with two double points.
Table 5.1 The triangular form of Pythagoras’ theorem.
Table 5.1 The triangular form of Pythagoras’ theorem.
The inverse transformation is
The inverse transformation is
Fig. 5.7 The tetrahedral number Tet, = 20.
Fig. 5.7 The tetrahedral number Tet, = 20.
Fig. 5.10 The body-centred cubic number Bcc2 = 9.
Fig. 5.10 The body-centred cubic number Bcc2 = 9.
Fig. 6.1 An integer parallelogram.
Fig. 6.1 An integer parallelogram.
Substituting these expressions back into Heron’s formula for [PCB], and using the relation (6.2.3) between [ABCD] and [PCB], we obtain Brahmagupta’s formula. In order to find a quadrilateral with integer sides and integer area we choose fourteen integer parameters and set Quadrilaterals and triangles
Substituting these expressions back into Heron’s formula for [PCB], and using the relation (6.2.3) between [ABCD] and [PCB], we obtain Brahmagupta’s formula. In order to find a quadrilateral with integer sides and integer area we choose fourteen integer parameters and set Quadrilaterals and triangles
Fig. 6.2 A cyclic quadrilateral with integer sides and integer area.
Fig. 6.2 A cyclic quadrilateral with integer sides and integer area.
Fig. 6.4 An isosceles trapezium with integer sides, integer diagonals, integer area, and integer circumradius.
Fig. 6.4 An isosceles trapezium with integer sides, integer diagonals, integer area, and integer circumradius.
Fig. 6.7 An equilateral triangle with integer side and a point P on the circumcircle with AP, BP, and CP integers.
Fig. 6.7 An equilateral triangle with integer side and a point P on the circumcircle with AP, BP, and CP integers.
Fig. 6.6 An equilateral triangle with integer side and a point P on BC with AP, BP, and CP integers.
Fig. 6.6 An equilateral triangle with integer side and a point P on BC with AP, BP, and CP integers.
Table 6.2 Integer-sided triangles with integer Fermat distances AF’, BF’, and CF. qa ad — AL TDL TUL Is all WICeer, A separate search to provide the least positive integer values of J, m, and n suc that all of m? + mn + n?, n? + nl +1’, and l? + lm +m? are perfect squares he also been carried out and reveals eight solutions with O <1 <m<n < 5016. TI results are shown in Table 6.2. This table provides integer-sided triangles with sides | b, and c which possess integer Fermat distances AF = 1], BF = m, and CF = n. also provides equilateral triangles of side d in which there is an internal point P wit AP =a, BP = b, and CP = c. Tables 6.1 and 6.2 are not independent, since tt problems they solve are in correspondence. However, Table 6.2 gives solutions wit
Table 6.2 Integer-sided triangles with integer Fermat distances AF’, BF’, and CF. qa ad — AL TDL TUL Is all WICeer, A separate search to provide the least positive integer values of J, m, and n suc that all of m? + mn + n?, n? + nl +1’, and l? + lm +m? are perfect squares he also been carried out and reveals eight solutions with O <1 <m<n < 5016. TI results are shown in Table 6.2. This table provides integer-sided triangles with sides | b, and c which possess integer Fermat distances AF = 1], BF = m, and CF = n. also provides equilateral triangles of side d in which there is an internal point P wit AP =a, BP = b, and CP = c. Tables 6.1 and 6.2 are not independent, since tt problems they solve are in correspondence. However, Table 6.2 gives solutions wit
Table 6.1 Equilateral triangles with integer side d and a point P at integer distance from its vertices.
Table 6.1 Equilateral triangles with integer side d and a point P at integer distance from its vertices.
Fig. 6.8 A triangle with integer sides and integer Fermat distances.
Fig. 6.8 A triangle with integer sides and integer Fermat distances.
Fig. 6.9 An equilateral triangle with integer side and an internal point P with AP, BP, and CP integers.
Fig. 6.9 An equilateral triangle with integer side and an internal point P with AP, BP, and CP integers.
Fig. 8.2 An integer-sided triangle with a pedal triangle having integer sides.
Fig. 8.2 An integer-sided triangle with a pedal triangle having integer sides.
Table 8.1 Integer values of 1, m, and n such that each of n? + nl + 17, 1? + lm + m?, anc m? +mn +n? are 3 times a perfect square.
Table 8.1 Integer values of 1, m, and n such that each of n? + nl + 17, 1? + lm + m?, anc m? +mn +n? are 3 times a perfect square.
Fig. 8.3 The Euler line and the nine-point circle DLEVNFUEMW.
Fig. 8.3 The Euler line and the nine-point circle DLEVNFUEMW.
Fig. 8.4 The analogue of the Euler line for the two incircles.
Fig. 8.4 The analogue of the Euler line for the two incircles.
fig. 9.2 An isosceles tetrahedron with integer edges and integer volume 7 950 571 200.
fig. 9.2 An isosceles tetrahedron with integer edges and integer volume 7 950 571 200.
Fig. 9.3 An isosceles tetrahedron with integer edges and integer circumradius.
Fig. 9.3 An isosceles tetrahedron with integer edges and integer circumradius.
Table 9.1 The five regular solids.
Table 9.1 The five regular solids.
Table 9.2 The six regular hypersolids.
Table 9.2 The six regular hypersolids.
Fig. 10.1 Three unit circles passing through O and the unit circle through their other points of intersection.
Fig. 10.1 Three unit circles passing through O and the unit circle through their other points of intersection.
Fig. 10.2 The Simson conic of Q and the Simson line of P with respect to Q.
Fig. 10.2 The Simson conic of Q and the Simson line of P with respect to Q.

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References (32)

  1. 4.3 As in Example 4.4.2 of the text, x = 1 (mod 4) and y is even. Now y 2 + 4 = (x + 3)(x 2 -3x + 9) and the quadratic is 3 (mod 4), and must therefore have a prime factor that is 3 (mod 4). However, y 2 + 4 can have no such factor. Here we are using the result that, if a 2 + b 2 has a factor k = 3 (mod 4), then k|a and k|b.
  2. 4.4 From the text, the condition is 2s|(3r 2 + a). Exercises 5.1
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  8. 3.1 (i) Truncated tetrahedron: 4 hexagons, 4 triangles, 12 vertices, and 18 edges. (ii) Truncated cube: 6 octagons, 8 triangles, 24 vertices, and 36 edges. (iii) Cuboctahedron: 6 squares, 8 triangles, 12 vertices, and 24 edges. (iv) Truncated octahedron: 8 hexagons, 6 squares, 24 vertices, and 36 edges. (v) Small rhombicuboctahedron: 16 squares, 8 triangles, 22 vertices, and 44 edges.
  9. Great rhombicuboctahedron: 6 octagons, 12 squares, 40 vertices, and 60 edges. (vii) Snub cube: 6 squares, 32 triangles, 24 vertices, and 60 edges. (viii) Icosidodecahedron: 12 pentagons, 20 triangles, 30 vertices, and 60 edges. (ix) Truncated dodecahedron: 12 decagons, 20 triangles, 60 vertices, and 90 edges.
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