===== n-gons for n to 8 =====
==== Legenda ====
**R** denotes the number of regions resulting from cutting convex polygon by all it's diagonals
**Ip** denotes the number of inner points in which diagonals of polygon are intersected
**si** denotes the number of inner points in which exactly **i+2** diagonals are intersected
R = {(n-1)(n-2)(n^2-3n+12)}/24 - 1/2 sum{}{}{k(k+1)s_k}
Ip = {(n-1)(n-2)(n-3)(n-4)}/24 - 1/2 sum{}{}{k(k+3)s_k}
----
==== Triangle ====
**R = 1**\\
**Ip = 0**
{{:marathon:3.png|triangle}}
----
==== Quadrangle ====
**R = 4**\\
**Ip = 1**
{{:marathon:4.png|quadrangle}}
----
==== Pentagon ====
**R = 10+0+1 = 11**\\
**Ip = 5**
{{:marathon:5.png|pentagon}}
----
==== Hexagons ====
**s1 = 0**\\
**R = 19+3+3 = 25**\\
**Ip = 15**
{{:marathon:6_0.png|hexagon_0}}
----
**s1 = 1**\\
**R = 18 + 6 = 24**\\
**Ip = 15 -1·2 = 13**
{{:marathon:6_1.png|hexagon_1}}
----
==== Heptagons ====
**s1 = 0**\\
**R = 35+7+7+0+1 = 50** (= 33+10+6+1+0 = 32+11+7=0+0 = 31+13+6+0+0)\\
**Ip = 35**\\
(coordinates: [35, 0], [80, 50], [60, 110], [0, 130], [-60, 110], [-80, 50], [-35, 0])
{{:marathon:7_0.png|heptagon_0}}
----
**s1 = 1**\\
**R = 34+9+5+1+0 = 49** (= 32+12+5+0+0 = 31+14+4+0+0 = 30+16+3+0+0)\\
**Ip = 35 - 1·2 = 33**\\
(coordinates: [35, 0], [80, 50], [60, 110], [0, 130], [-60, 110], [-80, 50], [-35, 0])
{{:marathon:7_1.png|heptagon_1}}
----
**s1 = 2**\\
**R = 33+11+4+0+0 = 48** (= 31+15+2+0+0)\\
**Ip = 35 - 2·2 = 31**\\
(coordinates: [120,0],[80,80],[10,95],[-80,80],[-100,0],[-80,-80],[80,-80])
{{:marathon:7_2.png|heptagon_2}}
----
**s1 = 3**\\
**R = 32+14+1+0+0 = 47**\\
**Ip = 35 - 3·2 = 29**\\
(coordinates: [100,0],[80,80],[0,100],[-80,80],[-100,0],[-80,-80],[80,-80])
{{:marathon:7_3.png|heptagon_3}}
----
==== Octagons ====
**s1 = 0, s2 = 0**\\
**R = 91**\\
**Ip = 70**\\
(coordinates: [120,20],[80,80],[-20,120],[-80,65],[-100,0],[-80,-80],[-5,-100],[80,-80])
{{:marathon:8_0_0.png|octagon_0_0}}
----
**s1 = 1, s2 = 0**\\
**R = 91-1 = 90**\\
**Ip = 70-1·2 = 68**\\
(coordinates: [120,20],[80,80],[-33,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])
{{:marathon:8_1_0.png|octagon_1_0}}
----
**s1 = 2, s2 = 0**\\
**R = 91-2 = 89**\\
**Ip = 70-2·2 = 66**\\
(coordinates: [120,30],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])
{{:marathon:8_2_0.png|octagon_2_0}}
----
**s1 = 3, s2 = 0**\\
**R = 91-3 = 88**\\
**Ip = 70-3·2 = 64**\\
(coordinates: [120,0],[13520/97, 10320/97],[0,100],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])
{{:marathon:8_3_0.png|octagon_3_0}}
----
**s1 = 4, s2 = 0**\\
**R = 91-4 = 87**\\
**Ip = 70-4·2 = 62**\\
(coordinates: [100,0],[5200/57, 2000/57],[0,100],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])
{{:marathon:8_4_0.png|octagon_4_0}}
----
**s1 = 5, s2 = 0**\\
**R = 91-5 = 86**\\
**Ip = 70-5·2 = 60**\\
(coordinates: [7275/65, -300/13],[48,48],[0,60],[-48,48],[-60,0],[-48,-48],[0,-60],[48,-48])
{{:marathon:8_5_0.png|octagon_5_0}}
----
**s1 = 6, s2 = 0**\\
**R = 91-6 = 85**\\
**Ip = 70-6·2 = 58**\\
(coordinates: [72,0],[11760/145,8880/145],[0,72],[-48,48],[-72,0],[-48,-48],[0,-72],[48,-48])
{{:marathon:8_6_0.png|octagon_6_0}}
----
**s1 = 7, s2 = 0**\\
**R = 91-7 = 84**\\
**Ip = 70-7·2 = 56**\\
(coordinates: [100,225/14],[90,45],[0,50],[-90,45],[-100,225/14],[-90,-45],[0,-475/9],[90,-45])
{{:marathon:8_7_0.png|octagon_7_0}}
----
**s1 = 8, s2 = 0**\\
**R = 91-8 = 83**\\
**Ip = 70-8·2 = 54**\\
(coordinates: [[35/9, -100], [70, -70], [115, 35/18], [70, 70], [35/9, 768/7], [-70, 70], [-1815/19, 35/18], [-70, -70])
{{:marathon:8_8_0.png|octagon_8_0}}
----
**s1 = 9, s2 = 0**\\
**R = 91-9 = 82**\\
**Ip = 70-9·2 = 52**\\
(coordinates: [0, -100], [70, -70], [985/9, 35/9], [70, 70], [0, 120], [-70, 70], [-985/9, 35/9], [-70, -70])
{{:marathon:8_9_0.png|octagon_9_0}}
----
**s1 = 0, s2 = 1**\\
**R = 91-1·3 = 88**\\
**Ip = 70-1·5 = 65**\\
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[100,-100])
{{:marathon:8_0_1.png|octagon_0_1}}
----
**s1 = 1, s2 = 1**\\
**R = 91-1-1·3 = 87**\\
**Ip = 70-1·2-1·5 = 63**\\
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-1800/23,-1800/23],[0,-100],[90,-90])
{{:marathon:8_1_1.png|octagon_1_1}}
----
**s1 = 2, s2 = 1**\\
**R = 91-2-1·3 = 86**\\
**Ip = 70-2·2-1·5 = 61**\\
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])
{{:marathon:8_2_1.png|octagon_2_1}}
----
**s1 = 3, s2 = 1**\\
**R = 91-3-1·3 = 85**\\
**Ip = 70-3·2-1·5 = 59**\\
(coordinates: [120,0],[80,80],[0,100],[-80,80],[-100,0],[-1200/17,-1200/17],[0,-100],[80,-80])
{{:marathon:8_3_1.png|octagon_3_1}}
----
**s1 = 4, s2 = 1**\\
**R = 91-4-1·3 = 84**\\
**Ip = 70-4·2-1·5 = 57**\\
(coordinates: [10,-20],[20,-10],[20,10],[10,20],[-10,20],[-20,10],[-20,-10],[-10,-20])
{{:marathon:8_4_1.png|octagon_4_1}}
----
**s1 = 6, s2 = 1**\\
**R = 91-6-1·3 = 82**\\
**Ip = 70-6·2-1·5 = 53**\\
(coordinates: [1900/9,0],[90,90],[0,100],[-90,90],[-100,0],[-90,-90],[0,-100],[90,-90])
{{:marathon:8_6_1.png|octagon_6_1}}
----
**s1 = 8, s2 = 1**\\
**R = 91-8-1·3 = 80**\\
**Ip = 70-8·2-1·5 = 49**\\
(coordinates: [100,0],[1180/11,840/11],[20,60],[-310/13,420/13],[-75/2,0],[-1180/31,-840/31],[-280/19,-840/19],[155/4,-105/2])
{{:marathon:8_8_1.png|octagon_8_1}}
----