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Given a simplex and point, convert back and forth between areal and Cartesian coordinates
ResourceFunction["Areal"][simplex,point] converts point between Cartesian and areal coordinates based on simplex. 
Vertices of a 345 triangle give an example of a 2simplex. We can convert back and forth:
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Given a line segment or 1simplex defined by two points, some other points on the line can be calculated by areal coordinates:
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Areal coordinates always have a sum of one:
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The minimal sign of an areal coordinate determines if a point is inside, outside or on the boundary of a given simplex:
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By Pick’s theorem, interior+boundary/21=area, this triangle should have area 12+6/21=14:
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Some graphics primitives that correspond to simplices are also supported:
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The following are equivalent:
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The areal coordinates of the incenter are proportional to the sides of the triangle:
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The trilinear coordinate formula of a triangle center can be used with areal coordinates to generate that center; here is a calculation of the orthocenter, where the altitudes of the triangle intersect:
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A tetrahedron is an example of a 3simplex. We can convert back and forth:
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The areal coordinates correspond to the area/volume when the Cartesian point is substituted for a point of the simplex and divided by the original volume; if the point is outside of the simplex, a negative value is returned:
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The plane perpendicular to vector (1/3,1/3,1/3) has all totals equal to 1, so points on this plane are assumed to be areal coordinates and get converted to Cartesian coordinates. There exists a triangle where the areal and Cartesian coordinates are the same, but in general, there is no back function for this special case:
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Arrange 25 trees in 18 lines of five trees:
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Arrange 149 points in 241 lines of five:
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Wolfram Language 11.3 (March 2018) or above
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