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Ceva's Theorem

The Theorem

PlaneGeometry.Theorems.Ceva — Module

Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E and F respectively. (The segments AD, BE, and CF are known as cevians.) Then, using signed lengths of segments,

\[BD \times CE \times AF = DC \times EA \times FB.\]
source

ceva-plot-1.svg

Finding the ceva and medians

function ceva(tri, O)
    A, B, C = vertices(tri)
    AB, BC, CA = edges(tri)
    AO, BO, CO = [Edge(pt, O) for pt in vertices(tri)]
    D, E, F = [concurrent(epair...) for epair in zip([BC, CA, AB], [AO, BO, CO])]

    # This mean no such points can be found. (This is possible)
    if D == nothing || E == nothing || F == nothing
        return nothing, nothing
    end

    elist = [Edge(pts...) for pts in zip([B, D, C, E, A, F], [D, C, E, A, F, B])]
    ([A, B, C, D, E, F], elist)
end

Examples

A = Point(0,0); B = Point(1, 3); C = Point(4,2); O = Point(2, 2)
plt, hold = ceva_draw(Triangle(A, B, C), O)
does_thmhold(hold)
Theorem holds! 😀️

ceva-plot-1.svg

plt, hold = ceva_rand()
does_thmhold(hold)
Theorem holds! 😀️

ceva-plot-2.svg

plt, hold = ceva_rand()
does_thmhold(hold)
Theorem holds! 😀️

ceva-plot-3.svg

Proof

function ceva_proof()
    @vars by cx positive=true
    @vars cy ox oy

    A = Point(0, 0); B = Point(0, by); C = Point(cx, cy); O = Point(ox, oy)
    tri = Triangle(A, B, C)

    plist, elist = ceva(tri, O)
    ceva_check(elist...)
end
does_thmhold(ceva_proof())
Theorem holds! 😀️