Teaching Kids Programming – Butterfly Theorem in Quadrilateral (Geometry)

ACMer

2 years ago

Teaching Kids Programming: Videos on Data Structures and Algorithms

Convex and Concave Quadrilateral

A quadrilateral is a polygon with four sides and four vertices. The distinction between concave and convex quadrilaterals lies in the arrangement of their angles and vertices:

Convex Quadrilateral

Definition: A quadrilateral is convex if all its interior angles are less than 180 degree, and no line segment between two vertices lies outside the shape.

    A-------B
   /         \
  D-----------C

Key Properties:
All vertices point outward.
The diagonals lie entirely inside the quadrilateral.

If you pick any two points within the quadrilateral, the line segment connecting them will always lie inside the quadrilateral.
Example: Squares, rectangles, parallelograms, and rhombuses are all convex quadrilaterals.

Concave Quadrilateral

Definition: A quadrilateral is concave if at least one interior angle is greater than 180 degree , causing the shape to “cave in.”

    A    D___E
   / \___/   \
  D-----------C

Key Properties:
At least one vertex points inward.
At least one diagonal lies partly or entirely outside the quadrilateral.

If you pick certain pairs of points within the quadrilateral, the line segment connecting them may pass outside the shape.
Example: A Dart, Arrowhead-shaped quadrilaterals and certain irregular shapes can be concave.

Butterfly Theorem in Quadrilateral (Geometry)

Given a convex Quadrilateral ABCD where the diagonals intersect at E, see below

     A-------B
     |\  S1 /|
     | \   / |
     |  \ /  |
     |S2 E S4|
     |  / \  |
     | /   \ |
     |/ S3  \|
     D-------C

We have:

where is the height from A to BD perpendicular.
Thus,
.

Similarly,
For and ,

where is the height from C to BD (perpendicular).
Thus,

From the above relationships, it follows that:

So, we have:

The Butterfly Theorem states that when diagonals of a convex quadrilateral intersect at a point E, the ratios of the areas of triangles formed by the diagonals and opposite sides are equal.

Proof Continuation

Now consider the sum of the areas of triangles and :

and the sum of and :

Dividing by , we get:

Simplifying, the terms and cancel out, leaving:

The sum of the areas and over and is equal to the ratio of the lengths of the segments and :

Thus, the Butterfly Theorem is verified, showing that the area and segment ratios align perfectly within the geometry of the intersecting diagonals.

Similarly, we would have: