British Flag Theorem: A Symmetry in Rectangular Geometry

In Euclidean geometry, the British Flag Theorem reveals a surprising symmetry involving distances from a point to the corners of a rectangle. This elegant result connects algebra and geometry, making it a useful identity in coordinate geometry, problem solving, and computer graphics. In this blog post, we will define the British Flag Theorem, prove it using coordinate geometry, verify it with Python, and explore its applications.

https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/British_flag_theorem_equal_areas.svg/330px-British_flag_theorem_equal_areas.svg.png

According to the British flag theorem, the red squares have the same total area as the blue squares

1. Definition

The British Flag Theorem is a geometric identity that applies to any rectangle. It states:

If a point $P$ lies anywhere inside or on the boundary of a rectangle with vertices $A$ , $B$ , $C$ , and $D$ , then the following relationship holds:

$PA^2 + PC^2 = PB^2 + PD^2$
Here, $PA$ , $PB$ , $PC$ , and $PD$ represent the distances from point $P$ to each of the four vertices of the rectangle. This identity showcases symmetry with respect to the rectangle’s diagonals.

2. Coordinate Geometry Proof

We can prove the British Flag Theorem using coordinate geometry.

Let the rectangle have vertices:
- $A = (0, 0)$
- $B = (a, 0)$
- $C = (a, b)$
- $D = (0, b)$
Let $P = (x, y)$ be any point in or on the rectangle.
Compute the squared distances:
- $PA^2 = x^2 + y^2$
- $PC^2 = (x - a)^2 + (y - b)^2$
- $PB^2 = (x - a)^2 + y^2$
- $PD^2 = x^2 + (y - b)^2$
Add both sides:
- Left: $PA^2 + PC^2 = x^2 + y^2 + (x - a)^2 + (y - b)^2$
- Right: $PB^2 + PD^2 = (x - a)^2 + y^2 + x^2 + (y - b)^2$
Since both expressions are algebraically equivalent, the identity holds.

3. Python Verification

We can verify the British Flag Theorem numerically using Python.

def squared_distance(p1, p2):
    return (p1[0] - p2[0])**2 + (p1[1] - p2[1])**2
 
def british_flag_theorem(rectangle, point):
    A, B, C, D = rectangle
    PA2 = squared_distance(point, A)
    PC2 = squared_distance(point, C)
    PB2 = squared_distance(point, B)
    PD2 = squared_distance(point, D)
 
    left = PA2 + PC2
    right = PB2 + PD2
 
    print("PA² + PC² = " + str(left))
    print("PB² + PD² = " + str(right))
    return abs(left - right) < 1e-9
 
# Define rectangle vertices
A = (0, 0)
B = (4, 0)
C = (4, 3)
D = (0, 3)
 
# Define a point inside the rectangle
P = (2, 1)
 
# Verify the theorem
print("British Flag Theorem holds:", british_flag_theorem([A, B, C, D], P))

def squared_distance(p1, p2):
    return (p1[0] - p2[0])**2 + (p1[1] - p2[1])**2

def british_flag_theorem(rectangle, point):
    A, B, C, D = rectangle
    PA2 = squared_distance(point, A)
    PC2 = squared_distance(point, C)
    PB2 = squared_distance(point, B)
    PD2 = squared_distance(point, D)

    left = PA2 + PC2
    right = PB2 + PD2

    print("PA² + PC² = " + str(left))
    print("PB² + PD² = " + str(right))
    return abs(left - right) < 1e-9

# Define rectangle vertices
A = (0, 0)
B = (4, 0)
C = (4, 3)
D = (0, 3)

# Define a point inside the rectangle
P = (2, 1)

# Verify the theorem
print("British Flag Theorem holds:", british_flag_theorem([A, B, C, D], P))

4. Applications of the British Flag Theorem

The British Flag Theorem has useful applications in several domains:

Coordinate geometry: Helpful in proving distance identities and symmetry properties.
Computer graphics: Used in rendering algorithms and layout optimizations involving rectangular regions.
Mathematical problem solving: Frequently appears in geometry contests and Olympiads.
Educational demonstrations: A great example of using algebraic distance formulas to uncover geometric patterns.

5. Example: Verifying the Theorem with Specific Coordinates

Let’s verify the British Flag Theorem using a specific rectangle and interior point.

Rectangle vertices:
- $A = (0, 0)$
- $B = (4, 0)$
- $C = (4, 3)$
- $D = (0, 3)$
Point inside the rectangle: $P = (2, 1)$
Compute squared distances:
- $PA^2 = 2^2 + 1^2 = 4 + 1 = 5$
- $PC^2 = (2 - 4)^2 + (1 - 3)^2 = 4 + 4 = 8$
- $PB^2 = (2 - 4)^2 + 1^2 = 4 + 1 = 5$
- $PD^2 = 2^2 + (1 - 3)^2 = 4 + 4 = 8$
Compare:
- $PA^2 + PC^2 = 5 + 8 = 13$
- $PB^2 + PD^2 = 5 + 8 = 13$
Result: Both sides are equal, confirming the theorem.

6. British Flag Theorem in Higher Dimensions

The British Flag Theorem applies strictly to rectangles in two-dimensional Euclidean space. It does not extend directly to other polygons or to higher dimensions. However, the use of squared distances to analyze symmetry is common in 3D geometry and linear algebra.

7. Limitations of the British Flag Theorem

Despite its elegance, the theorem has several limitations:

Applies only to rectangles: The identity does not hold for general quadrilaterals or parallelograms.
Requires Euclidean space: It depends on standard distance formulas, and is not valid in non-Euclidean geometries.
No direct generalization: The symmetry is specific to right angles and equal opposite sides.

8. Related Concepts

Pythagorean Theorem: The basis for calculating squared distances between points.
Distance formula: Essential for expressing distances algebraically in coordinate geometry.
Rectangle diagonals: The theorem reflects balance across the diagonals of the shape.
Centroids and symmetry: Related squared-distance properties also arise in centroids of shapes.

9. References

Pi in the Sky