# Root3 Harmonics

## harmonic pencil within a circumscribed equilateral triangle projects the golden ratio

No. Name Description Value Caption
1 Point A Point on xAxis A = (-0.5, 0)
2 Point B Point on xAxis B = (0.5, 0)
3 Button fit   fit
4 Line a Line A, B a: y = 0
5 Circle c Circle through B with center A c: (x + 0.5)² + y² = 1
6 Circle d Circle through A with center B d: (x - 0.5)² + y² = 1
7 Point D Intersection point of c, d D = (0, 0.86603)
8 Point C Intersection point of c, d C = (0, -0.86603)
9 Line b Line D, C b: x = 0
10 Point E Intersection of a and b E = (0, 0)
11 Point F Intersection of c and a F = (-1.5, 0)
12 Point G Intersection of d and a G = (1.5, 0)
13 Line f Line A, D f: -0.86603x + 0.5y = 0.43301
14 Line g Line B, D g: -0.86603x - 0.5y = -0.43301
15 Line h Line A, C h: 0.86603x + 0.5y = -0.43301
16 Line i Line B, C i: 0.86603x - 0.5y = 0.43301
17 Point H Intersection of c and h H = (-1, 0.86603)
18 Point I Intersection of d and i I = (1, 0.86603)
19 Point J Intersection of c and f J = (-1, -0.86603)
20 Point K Intersection of d and g K = (1, -0.86603)
21 Line j Line J, K j: y = -0.86603
22 Triangle t1 Polygon D, J, K t1 = 1.73205
23 Segment t1b Segment D, J t1b = 2
24 Segment t1c Segment J, K t1c = 2
25 Segment t1a Segment K, D t1a = 2
26 Line k Line H, I k: y = 0.86603
27 Line l Line J, B l: -0.86603x + 1.5y = -0.43301
28 Line m Line K, A m: -0.86603x - 1.5y = 0.43301
29 Point L Intersection of l and m L = (0, -0.28868)
30 Circle e Circle through K with center L e: x² + (y + 0.28868)² = 1.33333
31 Point M Intersection point of e, a M = (-1.11803, 0)
32 Point N Intersection point of e, a N = (1.11803, 0)
33 Point O Intersection point of e, h O = (-0.80902, 0.53523)
34 Point P Intersection point of e, h P = (0.30902, -1.40126)
35 Point Q Intersection point of e, i Q = (0.80902, 0.53523)
36 Point R Intersection point of e, i R = (-0.30902, -1.40126)
37 Segment g000a1 sectionABA(M, A, B, N) g000a1 = 0.61803 a
38 Segment g000b sectionABA(M, A, B, N) g000b = 1 b
39 Segment g000a2 sectionABA(M, A, B, N) g000a2 = 0.61803 a
40 Number g000ab sectionABA(M, A, B, N) g000ab = 0.61803
41 Number g000ba sectionABA(M, A, B, N) g000ba = 1.61803
42 Line n Line D, O n: 0.33079x - 0.80902y = -0.70063
43 Line p Line D, M p: 0.86603x - 1.11803y = -0.96825
44 Line q Line D, R q: 2.26728x - 0.30902y = -0.26762
45 Line r Line D, P r: 2.26728x + 0.30902y = 0.26762
46 Line s Line D, N s: 0.86603x + 1.11803y = 0.96825
47 Line t Line D, Q t: 0.33079x + 0.80902y = 0.70063
48 Polygon y9 Polygon D, O, M, J, R, P, K, N, Q y9 = 3.83394
49 Segment y9a Segment D, O y9a = 0.87403
50 Segment y9b Segment O, M y9b = 0.61803
51 Segment y9c Segment M, J y9c = 0.87403
52 Segment y9d Segment J, R y9d = 0.87403
53 Segment y9e Segment R, P y9e = 0.61803
54 Segment y9f Segment P, K y9f = 0.87403
55 Segment y9g Segment K, N y9g = 0.87403
56 Segment y8h Segment N, Q y8h = 0.61803
57 Segment y9i Segment Q, D y9i = 0.87403
58 Segment cha1 Segment D, O cha1 = 0.87403 chord
59 Segment chb1 Segment D, M chb1 = 1.41421 chord
60 Segment chc1 Segment D, J chc1 = 2 chord
61 Segment chd1 Segment D, R chd1 = 2.28825 chord
62 Segment chd2 Segment D, P chd2 = 2.28825 chord
63 Segment chc2 Segment D, K chc2 = 2 chord
64 Segment chb2 Segment D, N chb2 = 1.41421 chord
65 Segment cha2 Segment D, Q cha2 = 0.87403 chord
66 Segment gc001a1 sectionAB(O, D, M) gc001a1 = 0.87403 a
67 Segment gc001b1 sectionAB(O, D, M) gc001b1 = 1.41421 b
68 Number gc001ab1 sectionAB(O, D, M) gc001ab1 = 0.61803
69 Segment gc001a2 sectionAB(Q, D, N) gc001a2 = 0.87403 a
70 Segment gc001b2 sectionAB(Q, D, N) gc001b2 = 1.41421 b
71 Number gc001ab sectionAB(Q, D, N) gc001ab = 0.61803
72 Segment gc002a1 sectionAB(M, D, R) gc002a1 = 1.41421
73 Segment gc002b1 sectionAB(M, D, R) gc002b1 = 2.28825 a+b
74 Number gc0021 sectionAB(M, D, R) gc0021 = 0.61803
75 Segment gc002a2 sectionAB(N, D, P) gc002a2 = 1.41421
76 Segment gc002b2 sectionAB(N, D, P) gc002b2 = 2.28825 a+b
77 Number gc0022 sectionAB(N, D, P) gc0022 = 0.61803
78 Point S Intersection of j and n S = (-4.23607, -0.86603)
79 Point T Intersection of j and p T = (-2.23607, -0.86603)
80 Point U Intersection of q and j U = (-0.23607, -0.86603)
81 Point V Intersection of r and j V = (0.23607, -0.86603)
82 Point Z Intersection of s and j Z = (2.23607, -0.86603)
83 Point A1 Intersection of t and j A1 = (4.23607, -0.86603)
84 Segment g001c1 sectionAB(J, T, S) g001c1 = 1.23607 a+b
85 Segment g001d1 sectionAB(J, T, S) g001d1 = 2 a+2b
86 Number g001cd1 sectionAB(J, T, S) g001cd1 = 0.61803
87 Segment g001c2 sectionAB(K, Z, A1) g001c2 = 1.23607 a+b
88 Segment g001d2 sectionAB(K, Z, A1) g001d2 = 2 a+2b
89 Number g001cd2 sectionAB(K, Z, A1) g001cd2 = 0.61803
90 Segment g001b1 sectionBAB(J, U, V, K) g001b1 = 0.76393 b
91 Segment g001a sectionBAB(J, U, V, K) g001a = 0.47214 a
92 Segment g001b2 sectionBAB(J, U, V, K) g001b2 = 0.76393 b
93 Number g001ab2 sectionBAB(J, U, V, K) g001ab2 = 1.61803
94 Number g001ab1 sectionBAB(J, U, V, K) g001ab1 = 0.61803
95 Point W Intersection of i and p W = (1.80902, 2.26728)
96 Point B1 Intersection of i and n B1 = (1.30902, 1.40126)
97 Point C1 Intersection of i and s C1 = (0.69098, 0.33079)
98 Point D1 Intersection of r and i D1 = (0.19098, -0.53523)
99 Segment g002c1 sectionAB(B, D1, R) g002c1 = 0.61803 a+b
100 Segment g002d1 sectionAB(B, D1, R) g002d1 = 1 a+2b
101 Number g002cd1 sectionAB(B, D1, R) g002cd1 = 0.61803
102 Segment g002c2 sectionAB(I, B1, W) g002c2 = 0.61803 a+b
103 Segment g002d2 sectionAB(I, B1, W) g002d2 = 1 a+2b
104 Number g002cd2 sectionAB(I, B1, W) g002cd2 = 0.61803
105 Segment g002b1 sectionBAB(B, C1, Q, I) g002b1 = 0.38197 b
106 Segment g002a sectionBAB(B, C1, Q, I) g002a = 0.23607 a
107 Segment g002b2 sectionBAB(B, C1, Q, I) g002b2 = 0.38197 b
108 Number g002ab1 sectionBAB(B, C1, Q, I) g002ab1 = 1.61803
109 Number g002ab2 sectionBAB(B, C1, Q, I) g002ab2 = 0.61803
110 Point E1 Intersection of q and l E1 = (-0.17082, -0.3873)
111 Point F1 Intersection of r and l F1 = (0.1459, -0.20444)
112 Point G1 Intersection of s and l G1 = (0.8541, 0.20444)
113 Point H1 Intersection of t and l H1 = (1.17082, 0.3873)
114 Point I1 Intersection of k and l I1 = (2, 0.86603)
115 Segment g003a1 sectionABA(J, E1, H1, I1) g003a1 = 0.95745 a
116 Segment g003b sectionABA(J, E1, H1, I1) g003b = 1.54919 b
117 Segment g003a2 sectionABA(J, E1, H1, I1) g003a2 = 0.95745 a
118 Number g003ba2 sectionABA(J, E1, H1, I1) g003ba2 = 0.61803
119 Number g003ba1 sectionABA(J, E1, H1, I1) g003ba1 = 1.61803
120 Segment g004b1 sectionBAB(J, F1, G1, I1) g004b1 = 1.32317 b
121 Segment g004a sectionBAB(J, F1, G1, I1) g004a = 0.81776 a
122 Segment g004b2 sectionBAB(J, F1, G1, I1) g004b2 = 1.32317 b
123 Number g004ab sectionBAB(J, F1, G1, I1) g004ab = 1.61803
124 Number g004ba sectionBAB(J, F1, G1, I1) g004ba = 0.61803
125 Point J1 Intersection of p and l J1 = (-5.8541, -3.66854)
126 Point K1 Intersection of n and l K1 = (6.8541, 3.66854)
127 Segment g005b1 sectionBAB(J1, J, I1, K1) g005b1 = 5.60503 b
128 Segment g005a sectionBAB(J1, J, I1, K1) g005a = 3.4641 a
129 Segment g005b2 sectionBAB(J1, J, I1, K1) g005b2 = 5.60503 b
130 Number g005ab sectionBAB(J1, J, I1, K1) g005ab = 1.61803
131 Number g005ba sectionBAB(J1, J, I1, K1) g005ba = 0.61803
132 Line m1 Line R, M m1: -1.40126x - 0.80902y = 1.56665
133 Line n1 Line N, P n1: 1.40126x - 0.80902y = 1.56665
134 Line p1 Line O, N p1: 0.53523x + 1.92705y = 0.59841
135 Line q1 Line Q, M q1: 0.53523x - 1.92705y = -0.59841
136 Segment g006c sectionBAB(S, M, Q, I1) g006c = 3.23607 a+b
137 Segment g006b sectionBAB(S, M, Q, I1) g006b = 2 b
138 Segment g006a sectionBAB(S, M, Q, I1) g006a = 1.23607 a
139 Number g006ab sectionBAB(S, M, Q, I1) g006ab = 1.61803
140 Number g006bc sectionBAB(S, M, Q, I1) g006bc = 1.61803
141 Point L1 Intersection of r and q1 L1 = (0.07295, 0.33079)
142 Segment g007b1 sectionBAB(M, L1, Q, I1) g007b1 = 1.23607 b
143 Segment g007a sectionBAB(M, L1, Q, I1) g007a = 0.76393 a
144 Segment g007b2 sectionBAB(M, L1, Q, I1) g007b2 = 1.23607 b
145 Number o sectionBAB(M, L1, Q, I1) o = 1.61803
146 Number u sectionBAB(M, L1, Q, I1) u = 0.61803
147 Point M1 Intersection of f and q1 M1 = (-0.38197, 0.20444)
148 Point N1 Intersection of n1 and q1 N1 = (1.54508, 0.73967)
149 Segment g008a2 sectionABA(M, M1, Q, N1) g008a2 = 0.76393 a
150 Segment g008b sectionABA(M, M1, Q, N1) g008b = 1.23607 b
151 Segment g008a1 sectionABA(M, M1, Q, N1) g008a1 = 0.76393 a
152 Number g008ba sectionABA(M, M1, Q, N1) g008ba = 0.61803
153 Number g008ab sectionABA(M, M1, Q, N1) g008ab = 1.61803
154 Point O1 Intersection of n and a O1 = (-2.11803, 0)
155 Segment g009a sectionBAB(S, O1, B1, K1) g009a = 2.28825 a
156 Segment g009b sectionBAB(S, O1, B1, K1) g009b = 3.70246 b
157 Segment g009c sectionBAB(S, O1, B1, K1) g009c = 5.9907 a+b
158 Number g009bc sectionBAB(S, O1, B1, K1) g009bc = 0.61803
159 Number g009ab sectionBAB(S, O1, B1, K1) g009ab = 0.61803
160 Line r1 Line J, M r1: -0.86603x - 0.11803y = 0.96825
161 Line s1 Line J, O s1: -1.40126x + 0.19098y = 1.23586
162 Point P1 Intersection of n1 and p P1 = (2.92705, 3.13331)
163 Segment g010a1 sectionABA(M, D, W, P1) g010a1 = 1.41421 a
164 Segment g010b sectionABA(M, D, W, P1) g010b = 2.28825 b
165 Segment g010a2 sectionABA(M, D, W, P1) g010a2 = 1.41421 a
166 Number g010ba sectionABA(M, D, W, P1) g010ba = 0.61803
167 Number g010ab sectionABA(M, D, W, P1) g010ab = 1.61803
168 Line t1 Line J, Q t1: -1.40126x + 1.80902y = -0.1654
169 Line a1 Line J, N a1: -0.86603x + 2.11803y = -0.96825
170 Line b1 Line J, P b1: 0.53523x + 1.30902y = -1.66888
171 Line c1 Line J, R c1: 0.53523x + 0.69098y = -1.13364
172 Line d1 Line K, N d1: -0.86603x + 0.11803y = -0.96825
173 Line e1 Line K, Q e1: -1.40126x - 0.19098y = -1.23586
174 Line f2 Line K, O f2: -1.40126x - 1.80902y = 0.1654
175 Line g2 Line K, M g2: -0.86603x - 2.11803y = 0.96825
176 Line h2 Line K, R h2: 0.53523x - 1.30902y = 1.66888
177 Line i2 Line K, P i2: 0.53523x - 0.69098y = 1.13364