Centroid Exists Theorem

Centroid Exists Theorem

The Theorem
Finding the centroid and medians
Examples
Proof

The Theorem

PlaneGeometry.Theorems.Centroid — Module

The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The Centroids Exists Theorem states that the three medians indeed intersect at single point.

source

Finding the centroid and medians

PlaneGeometry.Theorems.Centroid.centroid — Function

centroid(📐️)

Find the centroid and the three medians of 📐️.

source

function centroid(tri)
    medians = map(i->median(circshift(vertices(tri), i)...), 0:2)
    cpt = concurrent(medians)
    cpt, medians 
end

Examples

PlaneGeometry.Theorems.Centroid.centroid_draw — Method

centroid_draw(📐️::Triangle)

Verify Centroid Exists Theorem for the triangle 📐️.

source

A = Point(0,0); B = Point(1, 3); C = Point(4,2)
plt, hold = centroid_draw(A, B, C)
does_thmhold(hold)

Theorem holds! 😀️

PlaneGeometry.Theorems.Centroid.centroid_rand — Method

centroid_rand()

Verify Centroid Exists Theorem for a random triangle.

source

plt, hold = centroid_rand()
does_thmhold(hold)

Theorem holds! 😀️

plt, hold = centroid_rand()
does_thmhold(hold)

Theorem holds! 😀️

Proof

function centroid_proof()
    @vars by cx positive=true;
    @vars cy;

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

    cpt, medians = centroid(tri)
    cpt != nothing
end

does_thmhold(centroid_proof())

Theorem holds! 😀️