Add Centroid theorem

README.md

@@ -1,4 +1,8 @@
-# PlaneGeomety
+# PlaneGeomety.jl
 
-This is collection used in my [blog post](https://newptcai.github.io/category/math.html) on using
-Julia to prove some basic plane geometry theorems.
+![Napoleon's Theorem](docs/src/assets/PlaneGeometry.svg)
+
+This package originated from a [blog post](https://newptcai.github.io/category/math.html) I wrote on
+proving Napoleon's theorem with Julia. I have since cleaned up the code and add a [new
+theorem](https://newptcai.github.io/PlaneGeometry.jl/theorem/Centroid/). Please see [the document](https://newptcai.github.io/PlaneGeometry.jl/) for
+details.

docs/make.jl

@@ -18,6 +18,7 @@ makedocs(sitename="Plane Geometry in Julia",
                   [
                    "Summary" => "theorem/index.md",
                    "Napoleon" => "theorem/Napoleon.md",
+                   "Centroid" => "theorem/Centroid.md",
                   ],
                   "Utitlies" => "util.md",
                   "Ackowledgement" => "thanks.md",

docs/src/definition/distance.md

@@ -2,14 +2,18 @@
 
 ## Euclidean distance
 
+```@setup 1
+using PlaneGeometry, Plots
+using PlaneGeometry: A, B, C
+```
+
 ```@docs
 distance
 ```
 
 ### Source Code
 
-```@example code
-using PlaneGeometry # hide
+```@example 1
 @code_md distance(Point(), Point()) # hide
 ```
 
@@ -21,6 +25,53 @@ squaredist
 
 ### Source Code
 
-```@example code
+```@example 1
 @code_md squaredist(Point(), Point()) # hide
 ```
+
+## Midpoint of an edge
+
+```@docs
+midpoint
+```
+
+### Source Code
+
+```@example 1
+@code_md midpoint(Point(), Point()) # hide
+```
+### Picture
+
+```@example 1
+plot(Edge(A, B), leg=false, color=:orange)
+mid = midpoint(A, B)
+scatter!(shape([A, B, mid]), leg=false, aspect_ratio=:equal, 
+    color=[:red, :red, :green], series_annotations = text.(["A", "B", "mid"], :bottom))
+```
+
+## Concurrent Point
+
+```@docs
+concurrent
+```
+
+### Source Code
+
+```@example 1
+@code_md concurrent(Edge[]) # hide
+```
+### Picture
+
+```@example 1
+elist = map(i->median(circshift([A, B, C], i)...), 0:2)
+cpt = concurrent(elist)
+plot(Triangle(A, B, C), fill=(0, :green), aspect_ratio=:equal, fillalpha= 0.2)
+for e in elist
+    plot!(e, leg=false, color=:orange)
+end
+scatter!(shape([A, B, C, cpt]), 
+    leg=false,
+    aspect_ratio=:equal, 
+    color=[:red, :red, :red, :green], 
+    series_annotations = text.(["A", "B", "C", "cpt"], :bottom))
+```

docs/src/definition/triangle.md

@@ -92,3 +92,26 @@ for t in tri_out
 end
 plot!(tri, fill=(0, :green), aspect_ratio=:equal, fillalpha= 0.2)
 ```
+
+## Median
+
+```@docs
+median
+```
+
+### Source Code
+
+```@example 1
+@code_md median(Point(), Point(), Point()) # hide
+```
+
+### Picture
+
+```@example 1
+me = median(A, B, C)
+mid = midpoint(A, B)
+scatter(shape([A, B, C, mid]), leg=false, color=:red, 
+    series_annotations = text.(["A", "B", "C", "mid"], :bottom))
+plot!(me, color=:orange)
+plot!(tri, fill=(0, :green), aspect_ratio=:equal, fillalpha= 0.2)
+```

docs/src/theorem/Centroid.md

@@ -0,0 +1,73 @@
+# Centroid Exists Theorem
+
+See [Napoleon's Theorem](Napoleon.md) for an example with more explanations.
+
+```@setup 1
+using 
+    PlaneGeometry, 
+    PlaneGeometry.Theorems, 
+    PlaneGeometry.Theorems.Napoleon,
+    Plots
+```
+
+```@contents
+Pages = ["Centroid.md"]
+```
+
+## The Theorem
+
+```@docs
+PlaneGeometry.Theorems.Centroid
+```
+
+## Finding the centroid and medians
+
+```@docs
+centroid
+```
+```@example 1
+@code_md centroid(Triangle()) # hide
+```
+
+## Examples
+
+```@docs
+centroid_draw(::Triangle)
+```
+
+```@example 1
+A = Point(0,0); B = Point(1, 3); C = Point(4,2)
+plt, hold = centroid_draw(A, B, C)
+does_thmhold(hold)
+savefig("centroid-plot-1.svg"); nothing # hide
+```
+
+![centroid-plot-1.svg](centroid-plot-1.svg)
+
+```@docs
+centroid_rand()
+```
+
+```@example 1
+plt, hold = centroid_rand()
+does_thmhold(hold)
+savefig("centroid-plot-2.svg"); nothing # hide
+```
+![centroid-plot-2.svg](centroid-plot-2.svg)
+
+```@example 1
+plt, hold = centroid_rand()
+does_thmhold(hold)
+savefig("centroid-plot-3.svg"); nothing # hide
+```
+![centroid-plot-3.svg](centroid-plot-3.svg)
+
+## Proof
+
+```@example 1
+@code_md centroid_proof() # hide
+```
+
+```@example 1
+does_thmhold(centroid_proof())
+```

docs/src/theorem/Napoleon.md

@@ -26,7 +26,7 @@ Given a triangle, to find its Napoleon triangle, we use the following function.
 napoleon_tri
 ```
 ```@example 1
-@code_md napoleon_tri(Triangle()) #
+@code_md napoleon_tri(Triangle()) # hide
 ```
 
 ## Examples
@@ -93,7 +93,7 @@ symbols provided by `SymPy`. Then we can just use `napoleon_tri` to find its Nap
 check if it is equilateral. We put the code in the following function.
 
 ```@example 1
-@code_md napoleon_proof()
+@code_md napoleon_proof() # hide
 ```
 
 And the final proof is a simple as one line of code ✌️.

src/PlaneGeometry.jl

@@ -6,11 +6,11 @@ export
     GeoObject, Point, Edge, Triangle, Circle,
     vertices, edges,
 
-    # metrics
-    distance, squaredist,
+    # distances
+    distance, squaredist, midpoint, concurrent,
 
     # triangles
-    equipoints, outer_equitri, outer_equitriangles, isequilateral, 
+    equipoints, outer_equitri, outer_equitriangles, isequilateral, median,
 
     # circles
     circumcircle,
@@ -31,9 +31,20 @@ using .Theorems.Napoleon
 export 
     napoleon_proof, napoleon_draw, napoleon_rand, napoleon_tri
 
+using .Theorems.Centroid
+export 
+    centroid_proof, centroid_draw, centroid_rand, centroid
+
 using .GeoPlots
 export 
     plot, plot!, shape
 
+function __init__()
+    global A, B, C
+
+    A = Point(0, 0); 
+    B = Point(1, 3); 
+    C = Point(4, 2);
+end
 
 end # module

src/distance.jl

@@ -21,11 +21,8 @@ end
 """
     distance(A, B)
 
-Compute the euclidean distance between `A` and `B`.
-
-# Mathematical definition
-
-> Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points.
+Compute the euclidean distance between `A` and `B`, which is the "ordinary" straight-line distance
+between two points.
 
 # Examples
 ```jldoctest
@@ -36,3 +33,51 @@ julia> distance(Point(0, 0), Point(3,4))
 function distance(A, B)
     SymPy.sqrt(squaredist(A, B))
 end
+
+
+"""
+    midpoint(A, B)
+
+Find the midpoint between `A` and `B`.
+
+# Examples
+```jldoctest
+julia> midpoint(Point(0, 0), Point(2,4))
+Point(1, 2)
+```
+"""
+function midpoint(A, B)
+    x1 = (A.x + B.x)/2
+    y1 = (A.y + B.y)/2
+    Point(x1, y1)
+end
+
+
+"""
+    concurrent(edgelist::Vector{Edge})
+
+Find the point where all `edges` (lines) intersect if such point exists. Return nothing otherwise.
+"""
+function concurrent(edgelist::Vector{Edge})
+    enum = length(edgelist)
+    if enum == 1
+        throw(ArgumentError("At least two edges are needed. $enum is given."))
+    end
+
+    @vars x y
+
+    eqs = Sym[]
+    for e in edgelist
+        # This should be 0
+        eq = (x-e.src.x)*(y-e.dst.y) - (x-e.dst.x)*(y-e.src.y)
+        push!(eqs, eq)
+    end
+
+    sol = solve(eqs, [x,y])
+
+    if length(sol) == 0
+        return nothing
+    else
+        return Point(simplify(sol[x]), simplify(sol[y]))
+    end
+end

src/elementary.jl

@@ -1,4 +1,5 @@
 import Base: isequal, ==, show
+using SymPy
 
 "A plane geometric object."
 abstract type GeoObject end
@@ -18,9 +19,11 @@ A = Point(0,0); B = Point(1, 3); C = Point(4,2)
 """
 struct Point <: GeoObject
     "x cooridnate"
-    x::Number
+    x::Sym
     "y cooridnate"
-    y::Number
+    y::Sym
+
+    Point(x, y) = new(Sym(x), Sym(y))
 end
 
 """
@@ -37,10 +40,6 @@ Point() = Point(0, 0)
 
 show(io::IO, pt::Point) = print(io, "Point($(pt.x), $(pt.y))")
 
-const A = Point(0,0); 
-const B = Point(1, 3); 
-const C = Point(4,2)
-
 """
     Triangle(A, B, C) 
 
@@ -109,6 +108,9 @@ struct Edge <: GeoShape
     dst::Point
 end
 
+"Check if two edges are at the same."
+(==)(e1::Edge, e2::Edge) = e1.src==e2.src && e1.dst==e2.dst
+
 "Get the list of edges of the triangle `tri`."
 function edges(tri::Triangle)
     elist = Edge[]

src/plots.jl

@@ -53,15 +53,34 @@ end;
 """
     plot(obj::GeoObject, args...; kwargs...)
 
-Plot a geometric object `obj`.
+Plot `obj` on current plot.
 """
 plot(obj::GeoObject, args...; kwargs...) = plot(shape(obj), args...; kwargs...)
 
 """
     plot!(obj::GeoObject, args...; kwargs...)
 
-Plot a geometric object `obj` on current plot.
+Plot `obj` on current plot.
 """
 plot!(obj::GeoObject, args...; kwargs...) = plot!(shape(obj), args...; kwargs...)
 
+"""
+    plot!(edge::Edge, args...; kwargs...)
+
+Plot `edge` on current plot.
+"""
+function plot!(edge::Edge, args...; kwargs...)
+    eshape = shape(edge)
+    plot!(eshape.x, eshape.y, args...; kwargs...)
+end
+
+"""
+    plot(edge::Edge, args...; kwargs...)
+
+Plot `edge` on current plot.
+"""
+function plot(edge::Edge, args...; kwargs...)
+    eshape = shape(edge)
+    plot(eshape.x, eshape.y, args...; kwargs...)
+end
 end