+1222 Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
An equilateral triangle, a regular dodecagon, and a regular -gon, all with the same side length, also completely surround a point. Find n.
+23 For polygons with a, b, and c sides to surround a point, 1/a + 1/b + 1/c should equal 1/2.
For example, two regular pentagons and a regular decagon surround a point because 1/5 + 1/5 + 1/10 = 2/10 + 2/10 + 1/10 = 5/10 = 1/2
For an equilateral triangle and a dodecagon,
1/3 + 1/12 + 1/n = 1/2
4/12 + 1/12 + 1/n = 1/2
5/12 + 1/n = 6/12
1/n = 1/12
n = 12
This means that an equilateral triangle and two regular dodecagons completely surround a point.
