perplexus.info :: Geometry : Ring of triangles II

Ring of triangles II (Posted on 2016-03-10)

As in Ring of triangles I, ABCDEF is a hexagon and O is an interior point. Triangles ABO, CDO, EFO are is equilateral. Their respective side lengths are different integers with no common divisor. (They are also not 4,3,7 because that was part I.)

BC, DE, and FA are also integer lengths.

Find a set of lengths {AB,BC,CD,DE,EF,FA} for which this is possible.

Note: There are sides of the hexagon that can be the same.

No Solution Yet Submitted by Jer

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computer solution

| Comment 1 of 2

Just had to add outer loops (ab, cd, ef) to my program for Ring of Triangles I, and change the lower and upper bounds for each side (bc, de, fa) to be calculated instead of hard coded:

DefDbl A-Z

Dim crlf$, pi

Private Sub Form_Load()

Form1.Visible = True

crlf = Chr(13) + Chr(10)

Text1.Text = ""

pi = 4 * Atn(1)

For ab = 1 To 25

For cd = ab + 1 To 26

If gcd(ab, cd) = 1 Then

For ef = cd + 1 To 27

DoEvents

If gcd(ab * cd, ef) = 1 Then

For bc = cd - ab + 1 To cd + ab - 1

For de = ef - cd + 1 To ef + cd - 1

For fa = ef - ab + 1 To ef + ab - 1

a1 = angle(bc, cd, ab)

a2 = angle(de, ef, cd)

a3 = angle(fa, ef, ab)

diff = Abs(180 - a1 - a2 - a3)

If diff < 0.000001 Then

Text1.Text = Text1.Text & ab & Str(cd) & Str(ef) & crlf

Text1.Text = Text1.Text & bc & Str(de) & Str(fa) & crlf

Text1.Text = Text1.Text & a1 & Str(a2) & Str(a3) & " " & Str(a1 + a2 + a3) & crlf & crlf

End If

Text1.Text = Text1.Text & crlf & " done"

End Sub

Function angle(a, b, c)

cosa = (b * b + c * c - a * a) / (2 * b * c)

If cosa = 0 Then angle = 90: Exit Function

sina = Sqr(1 - cosa * cosa)

a1 = Atn(sina / cosa) * 180 / pi

If cosa < 0 Then a1 = a1 + 180

angle = a1

End Function

Function gcd(a, b)

x = a: y = b

q = Int(x / y)

z = x - q * y

x = y: y = z

Loop Until z = 0

gcd = x

End Function

looks for cases in which the largest of the equilateral-triangle sides is no larger than 28. It looks for these to be in ascending order wlog.

The format of each set below is:

ab cd ef

bc de fa

angle boc, angle doe, angle foa total of angles

The total of the angles must be 180.

Ignore the second solution, as it is the one from part I.

Except for that second solution, the first five solutions repeat the values of ab, cd, and ef in the derived set bc, de and fa. The sixth solution is the first new one where the found values do not echo the given values.

Some can have more than one solution, such as 8, 19 and 25 for ab, cd and ef, resulting in either 15, 33, 19 or 8, 19, 25 for bc, de and fa.

3 4 5

5 3 4

90 36.869897645844 53.130102354156 180

3 4 7

3 7 6

48.1896851042214 73.3984504009798 58.4118644947988 180

3 5 7

7 3 5

120 21.7867892982618 38.2132107017382 180

3 7 8

8 3 7

98.2132107017382 21.7867892982618 60 180

3 10 11

11 3 10

101.536959032815 15.4987327565968 62.9643082105877 180

3 10 23

8 22 22

41.4096221092709 71.6260619966705 66.9643158940586 180

3 11 13

13 3 11

126.221546619835 10.7293735799286 43.0490798002363 180

3 13 14

14 3 13

103.342363797088 12.0345697280641 64.6230664748477 180

3 16 17

17 3 16

104.47751218593 9.83822698524653 65.6842608288235 180

3 17 19

19 3 17

128.14451310123 7.13324087398138 44.7222460247886 180

3 19 20

20 3 19

105.257523290456 8.32065518774547 66.4218215217982 180

3 22 23

23 3 22

105.826620131872 7.20906397406899 66.9643158940586 180

3 23 25

25 3 23

129.082207297359 5.34479670344638 45.5729959991943 180

3 25 26

26 3 25

106.260204708312 6.35966023972848 67.3801350519596 180

4 5 7

7 4 5

101.536959032815 34.0477323699915 44.415308597193 180

4 7 9

7 8 7

73.3984504009798 58.4118644947988 48.1896851042214 180

4 7 9

9 4 7

106.60154959902 25.2087652967584 48.1896851042214 180

4 9 11

11 4 9

109.471220634491 20.0499757241515 50.4788036413578 180

4 11 13

13 4 11

111.323686263498 16.6561861813541 52.020127555148 180

4 13 15

15 4 13

112.61986494804 14.2500326978036 53.130102354156 180

4 15 17

17 4 15

113.578178478202 12.4537005942687 53.9681209275294 180

4 17 19

19 4 17

114.315739171176 11.0608009807651 54.6234598480584 180

4 19 21

21 4 19

114.901062370367 9.94884220867952 55.1500954209535 180

4 21 23

23 4 21

115.376933525152 9.04045518528928 55.5826112895584 180

4 23 25

25 4 23

115.771461740551 8.28433600201726 55.9442022574321 180

4 25 27

27 4 25

116.10388113734 7.64510745854866 56.2510114041114 180

5 6 7

7 5 6

78.4630409671845 44.415308597193 57.1216504356225 180

5 7 8

8 5 7

81.7867892982618 38.2132107017382 60 180

5 7 9

9 5 7

95.7391704772668 33.5573097619207 50.7035197608125 180

5 7 11

11 5 7

132.177418600899 19.6850548246589 28.1375265744417 180

5 8 9

9 5 8

84.2608295227332 33.5573097619207 62.1818607153461 180

5 8 11

11 5 8

113.578178478202 24.6199773286571 41.8018441931411 180

5 9 11

11 5 9

99.5940682268605 26.6274783929746 53.7784533801649 180

5 9 13

10 10 10

86.1774462707257 50.1316584499894 43.690895279285 180

5 9 13

13 5 9

134.427004000806 15.9423686056286 29.6306273935657 180

5 9 17

11 13 13

99.5940682268605 48.938955479872 31.4669762932676 180

5 11 12

12 5 11

88.9582011495447 24.6199773286571 66.4218215217982 180

5 11 13

13 5 11

102.069082524351 22.0931769922725 55.8377404833764 180

5 11 14

14 5 11

117.035691789412 18.5489996133947 44.415308597193 180

5 11 17

9 21 13

53.7784533801649 94.7545703265675 31.4669762932676 180

5 12 13

13 5 12

90 22.6198649480404 67.3801350519596 180

5 12 17

9 21 14

42.8334280660673 91.1235186577004 46.0430532762323 180

5 13 14

14 5 13

90.881508306806 20.9222396825766 68.1962520106174 180

5 13 16

16 5 13

118.484635395177 15.9423686056286 45.5729959991943 180

5 13 17

17 5 13

136.950920199764 11.5821035069688 31.4669762932676 180

5 13 19

10 18 18

43.690895279285 65.3544154841787 70.9546892365363 180

5 14 17

17 5 14

119.059281068041 14.8976656557271 46.0430532762323 180

5 16 17

17 5 16

92.8659839825989 17.0825832434492 70.0514327739519 180

5 16 19

19 5 16

120 13.1735511072589 46.8264488927411 180

5 17 18

18 5 17

93.3722866834339 16.0989339510568 70.5287793655093 180

5 17 19

19 5 17

106.049670804167 14.6492229289679 59.3011062668647 180

5 17 21

21 5 17

138.336242374956 9.10682578759011 32.5569318374543 180

5 18 19

19 5 18

93.8225537292743 15.2227570341893 70.9546892365363 180

5 19 21

21 5 19

106.826448892741 13.1735511072589 60 180

5 19 22

22 5 19

121.050239116786 11.2276684820576 47.7220924011566 180

5 19 23

23 5 19

138.818850480643 8.22955879891473 32.9515907204427 180

5 21 22

22 5 21

94.9171003355288 13.087352664554 71.9955469999172 180

5 21 23

23 5 21

107.457603123722 11.9687455546856 60.5736513215923 180

5 22 23

23 5 22

95.2159085704541 12.5030233029529 72.2810681265929 180

5 23 24

24 5 23

95.4888575690365 11.9687455546856 72.5423968762779 180

5 23 26

26 5 23

122.034836044198 9.38247437392496 48.5826895818775 180

5 23 27

27 5 23

139.540916919211 6.90177331886874 33.5573097619207 180

5 26 27

27 5 26

96.1823042363935 10.6091428781952 73.2085528854113 180

6 7 11

7 12 7

64.6230664748477 80.2801322975677 35.0968012275846 180

6 7 11

11 6 7

115.376933525152 29.5262652472631 35.0968012275846 180

6 11 13

13 6 11

95.2159085704541 27.36306182235 57.4210296071959 180

6 11 17

13 12 13

95.2159085704541 44.6649245311407 40.1191668984051 180

6 13 17

11 20 13

57.4210296071959 82.459803494399 40.1191668984051 180

6 13 17

17 6 13

122.578970392804 17.3018627087907 40.1191668984051 180

6 17 19

17 12 17

79.8357513782765 38.4379622530885 61.726286368635 180

6 17 19

19 6 17

100.164248621723 18.1094650096416 61.726286368635 180

6 19 23

23 6 19

125.376540151942 12.2808543958512 42.3426054522072 180

6 23 25

25 6 23

102.555857798586 13.5480233387539 63.8961188626601 180

7 8 9

9 7 8

73.3984504009798 48.1896851042214 58.4118644947988 180

7 8 11

11 7 8

94.0960437581523 39.4005687537377 46.50338748811 180

7 8 13

13 7 8

120 27.795772496028 32.204227503972 180

7 9 10

10 7 9

76.2258530019733 42.8334280660673 60.9407189319595 180

7 9 11

11 7 9

85.9039562418477 39.4005687537377 54.6954750044146 180

7 9 13

13 7 9

108.030535393338 30.7983817102935 41.1710828963685 180

7 10 11

11 7 10

78.4630409671845 38.5726508222278 62.9643082105877 180

7 10 13

13 7 10

98.2132107017382 32.204227503972 49.5825617942898 180

7 11 12

12 7 11

80.2801322975677 35.0968012275846 64.6230664748477 180

7 11 13

13 7 11

89.6279468690078 32.5781984923066 57.7938546386856 180

7 11 15

15 7 11

110.924832427638 25.8419327631671 43.2332348091946 180

7 11 16

16 7 11

123.948089499638 21.2799664684131 34.7719440319486 180

7 11 17

17 7 11

140.599431246262 15.1509401512325 24.2496286025052 180

7 12 13

13 7 12

81.7867892982618 32.204227503972 66.0089831977662 180

7 12 17

17 7 12

124.849904579046 19.7499227956426 35.400172625311 180

7 13 15

15 7 13

92.204227503972 27.795772496028 60 180

7 13 16

16 7 13

102.051523401152 25.331675016713 52.6168015821351 180

7 13 17

17 7 13

112.961337256589 22.2804156886959 44.7582470547147 180

7 13 18

18 7 13

125.620762776558 18.4286696038241 35.9505676196178 180

7 13 19

19 7 13

141.786789298262 13.1735511072589 25.0396595944793 180

7 15 16

16 7 15

85.0828996644712 25.8419327631671 69.0751675723617 180

7 15 17

17 7 15

94.0960437581523 24.2496286025052 61.6543276393425 180

7 15 19

19 7 15

114.474342562847 19.5918322609051 45.9338251762476 180

7 16 17

17 7 16

85.9039562418477 24.2496286025052 69.8464151556471 180

7 16 19

19 7 16

104.47751218593 20.8990279660117 54.6234598480584 180

7 17 18

18 7 17

86.6277133165661 22.8435073179246 70.5287793655093 180

7 17 19

19 7 17

95.5456418654339 21.5120360095541 62.9423221250119 180

7 17 20

20 7 17

105.099981783404 19.7499227956426 55.1500954209535 180

7 17 22

22 7 17

127.839045349141 14.553269075053 37.6076855758059 180

7 17 23

23 7 17

143.371519783108 10.4618723347928 26.1666078820989 180

7 17 27

18 22 22

86.6277133165661 54.4298454144525 38.9424412689814 180

7 18 19

19 7 18

87.2705973632213 21.5925161781631 71.1368864586156 180

7 18 23

23 7 18

128.246619878345 13.8287435443043 37.9246365773508 180

7 19 20

20 7 19

87.8455156316498 20.4721827795487 71.6823015888015 180

7 19 22

22 7 19

106.152535317829 17.7955541818091 56.0519105003617 180

7 19 23

23 7 19

116.574969518987 15.7948290503887 47.6302014306242 180

7 19 24

24 7 19

128.613238191003 13.1735511072589 38.2132107017382 180

7 19 25

25 7 19

143.927314329076 9.48951461531528 26.5831710556088 180

7 20 23

23 7 20

106.60154959902 16.9574262941723 56.4410241068074 180

7 22 23

23 7 22

89.255878048784 17.717388023586 73.02673392763 180

7 22 25

25 7 22

107.379616807781 15.4987327565968 57.1216504356225 180

7 22 27

27 7 22

129.521196358642 11.5363623723765 38.9424412689814 180

7 23 24

24 7 23

89.6441233048479 16.9574262941723 73.3984504009798 180

7 23 25

25 7 23

98.3930328883628 16.0813096744844 65.5256574371528 180

7 23 26

26 7 23

107.718931873407 14.8600385193971 57.4210296071959 180

7 23 27

27 7 23

117.965777549082 13.2373216902555 48.7969007606629 180

7 24 25

25 7 24

90 16.260204708312 73.739795291688 180

7 25 26

26 7 25

90.3274062361796 15.6182367681917 74.0543569956287 180

7 25 27

27 7 25

99.0410961485752 14.8351115819757 66.1237922694491 180

7 26 27

27 7 26

90.6296366234414 15.0251850801672 74.3451782963914 180

8 9 11

11 8 9

80.4059317731395 45.8156148466955 53.7784533801649 180

8 9 11

13 5 9

99.5940682268605 26.6274783929746 53.7784533801649 180

8 9 13

11 11 9

80.4059317731395 56.5449884266241 43.0490798002363 180

8 9 13

13 8 9

99.5940682268605 37.3568519729032 43.0490798002363 180

8 9 17

11 16 11

80.4059317731395 68.1270919335929 31.4669762932676 180

8 9 17

13 13 11

99.5940682268605 48.938955479872 31.4669762932676 180

8 11 13

9 16 9

53.7784533801649 83.1724668195987 43.0490798002363 180

8 11 13

13 8 11

84.7840914295459 37.7948789632582 57.4210296071959 180

8 11 13

17 3 9

126.221546619835 10.7293735799286 43.0490798002363 180

8 11 15

15 8 11

103.136558786945 31.2904452138609 45.5729959991943 180

8 11 17

9 21 11

53.7784533801649 94.7545703265675 31.4669762932676 180

8 11 17

17 8 11

126.221546619835 22.3114770868973 31.4669762932676 180

8 11 19

11 20 13

68.6763137365021 78.6867887599442 32.6368975035537 180

8 11 23

5 20 25

24.6199773286571 60.3914806957868 94.9885419755561 180

8 13 15

15 8 13

87.795772496028 32.204227503972 60 180

8 13 17

9 24 11

43.0490798002363 105.483943906496 31.4669762932676 180

8 13 17

17 8 13

105.618498275649 26.949955927645 47.4315457967065 180

8 13 19

19 8 13

127.979872444852 19.3832300515943 32.6368975035537 180

8 13 21

6 14 26

18.7169506573713 40.6905605975357 120.592488745093 180

8 15 17

17 8 15

90 28.072486935853 61.927513064147 180

8 15 19

19 8 15

107.457603123722 23.6819072911771 48.8604895851009 180

8 15 23

11 26 19

45.5729959991943 83.509211298165 50.9177927026407 180

8 17 19

19 8 17

91.6854130402226 24.8895564787644 63.425030481013 180

8 17 21

21 8 17

108.876503630559 21.128697254301 49.9947991151398 180

8 17 23

23 8 17

130.320215069971 15.3786321732378 34.3011527567916 180

8 19 21

21 8 19

93.016961309816 22.3599722153363 64.6230664748477 180

8 19 23

23 8 19

110.005190088159 19.0770172091998 50.9177927026407 180

8 19 25

15 33 19

48.8604895851009 96.224304167455 34.9152062474442 180

8 19 25

25 8 19

131.139510414899 13.9452833376567 34.9152062474442 180

8 21 23

23 8 21

94.0960437581523 20.3001215245972 65.6038347172504 180

8 21 25

25 8 21

110.924832427638 17.3913020460274 51.6838655263343 180

8 23 25

25 8 23

94.9885419755561 18.5896365026458 66.4218215217982 180

8 23 27

27 8 23

111.688805618141 15.9810813461885 52.3301130356704 180

8 25 27

27 8 25

95.7391704772668 17.1462099988918 67.1146195238414 180

9 10 11

11 9 10

70.5287793655093 50.4788036413578 58.9924169931329 180

9 10 13

5 5 20

29.9264348666142 20.2052235833752 129.868341550011 180

9 10 13

13 9 10

86.1774462707257 43.690895279285 50.1316584499894 180

9 10 17

17 9 10

126.869897645844 25.057615418303 28.072486935853 180

9 10 19

11 11 20

70.5287793655093 26.5253520166097 82.945868617881 180

9 10 23

9 23 18

56.2510114041114 77.444142201414 46.3048463944746 180

9 11 13

5 11 17

26.6274783929746 53.7784533801649 99.5940682268605 180

9 11 13

5 18 11

26.6274783929746 96.8275331804013 56.5449884266241 180

9 11 13

8 18 8

45.8156148466955 96.8275331804013 37.3568519729032 180

9 11 13

12 16 6

72.9665898338858 83.1724668195987 23.8609433465154 180

9 11 13

13 9 11

80.4059317731395 43.0490798002363 56.5449884266241 180

9 11 14

4 11 19

20.0499757241515 50.4788036413578 109.471220634491 180

9 11 14

14 9 11

88.2634984240621 39.9831214542829 51.753380121655 180

9 11 16

8 13 17

45.8156148466955 53.7784533801649 80.4059317731395 180

9 11 16

16 9 11

105.826620131872 32.7637577588568 41.4096221092709 180

9 11 17

8 8 22

45.8156148466955 22.3114770868973 111.872908066407 180

9 11 17

17 9 11

116.065218482578 28.3963170018008 35.5384645156217 180

9 11 19

10 10 22

58.9924169931329 23.9534516247481 97.054131382119 180

9 11 19

19 9 11

143.420457162488 16.3967184923792 20.1828243451324 180

9 13 14

14 9 13

76.6576362029118 38.7192973222405 64.6230664748477 180

9 13 16

11 11 17

56.5449884266241 43.0490798002363 80.4059317731395 180

9 13 16

16 9 13

91.4692835813538 34.2160511312983 54.3146652873479 180

9 13 17

11 5 22

56.5449884266241 11.5821035069688 111.872908066407 180

9 13 17

17 9 13

99.5940682268605 31.4669762932676 48.938955479872 180

9 13 19

19 9 13

118.317621117008 24.6452747866058 37.0371040963865 180

9 13 20

20 9 13

129.868341550011 20.2052235833752 29.9264348666142 180

9 13 25

10 18 26

50.1316584499894 43.690895279285 86.1774462707257 180

9 14 17

17 9 14

92.7294026367787 31.9251664559684 55.3454309072529 180

9 14 19

19 9 14

109.471220634491 26.5253520166097 44.0034273488996 180

9 16 17

17 9 16

80.4059317731395 31.4669762932676 68.1270919335929 180

9 16 19

19 9 16

94.7801918471991 28.1665785968281 57.0532295559728 180

9 16 23

23 9 16

131.810314895779 16.9574262941723 31.2322588100491 180

9 17 19

19 9 17

88.3145869597774 28.2603825592097 63.425030481013 180

9 17 20

20 9 17

95.6262709461056 26.6046816893965 57.7690473644979 180

9 17 22

22 9 17

111.872908066407 22.3114770868973 45.8156148466955 180

9 17 23

23 9 17

121.305950066774 19.5319431376897 39.1621067955359 180

9 17 25

25 9 17

146.442690238079 11.4783409545336 22.0789688073871 180

9 19 20

20 9 19

82.945868617881 26.5253520166097 70.5287793655093 180

9 19 22

22 9 19

97.054131382119 23.9534516247481 58.9924169931329 180

9 19 23

23 9 19

104.737203743956 22.2365612708876 53.0262349851567 180

9 19 25

25 9 19

122.349860514935 17.7056342952502 39.9445051898149 180

9 19 26

26 9 19

133.173551107259 14.622221388769 32.204227503972 180

9 20 23

23 9 20

97.6622556607661 22.818330064839 59.5194142743949 180

9 22 23

23 9 22

84.7840914295459 22.9348404438612 72.2810681265929 180

9 22 25

25 9 22

98.7147420186181 20.8451663600604 60.4400916213215 180

9 23 25

25 9 23

92.0763885057186 21.0856802931633 66.837931201118 180

9 23 26

26 9 23

99.1732493837892 19.9821160424938 60.844634573717 180

9 25 26

26 9 25

86.1774462707257 20.2052235833752 73.6173301458992 180

10 11 13

13 10 11

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17 19 27

27 17 19

97.0243424609213 38.6751276497629 44.3005298893157 180

17 20 21

21 17 20

68.6105166738151 48.9176668595484 62.4718164666365 180

17 20 23

23 17 20

76.3910393692171 45.9207901521125 57.6881704786704 180

17 20 27

27 17 20

93.3722866834339 38.9424412689814 47.6852720475847 180

17 21 22

22 17 21

69.8464151556471 46.50338748811 63.6501973562429 180

17 21 23

14 34 14

41.5912771433651 101.101437361503 37.3072854951323 180

17 21 23

23 17 21

73.649497537451 45.1729648541994 61.1775376083496 180

17 21 23

32 16 10

114.315739171176 42.3426054522072 23.3416553766163 180

17 21 25

25 17 21

81.5434806647295 42.2685844295725 56.187934905698 180

17 21 26

26 17 21

85.6625640546941 40.6905605975357 53.6468753477701 180

17 22 23

23 17 22

70.9615130451116 44.3227781335677 64.7157088213207 180

17 22 25

25 17 22

78.5880982561865 41.8018441931411 59.6100575506724 180

17 22 27

27 17 22

86.6277133165661 38.9424412689814 54.4298454144525 180

17 23 24

16 19 29

44.0552116095984 47.6302014306242 88.3145869597774 180

17 23 24

24 17 23

71.9731337189693 42.3426054522072 65.6842608288235 180

17 23 25

25 17 23

75.7115854218448 41.2209585277303 63.0674560504249 180

17 23 26

26 17 23

79.5378653137259 40.0140953466979 60.4480393395762 180

17 23 27

27 17 23

83.464964788081 38.7217286208006 57.8133065911184 180

17 24 25

25 17 24

72.8953648233562

Posted by Charlie on 2016-03-10 09:43:35

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