**π‘ Problem Formulation:** When dealing with geometrical problems in programming, it’s common to encounter tasks such as finding the coordinates of the fourth vertex of a rectangle when three vertices are known. This task is especially relevant in computer graphics, computer vision and game development. For instance, given vertices A(2, 4), B(6, 4), and C(2, 8), we seek to determine the coordinates of the fourth vertex D. This article explores five Python methods to solve this problem.

## Method 1: Vector Addition and Subtraction

This method involves deducing the properties of a rectangle – that the diagonals bisect each other at the centre point. By finding the midpoint of the diagonal formed by two known points and performing vector operations, we can find the fourth vertex. The function `find_fourth_vertex()`

calculates the midpoint and uses vector addition and subtraction to find the missing coordinate.

Here’s an example:

def find_fourth_vertex(A, B, C): # Assuming A, B, C are given and A-B is one side and A-C is another side mid_point = ((B[0] + C[0]) / 2, (B[1] + C[1]) / 2) D = (2 * mid_point[0] - A[0], 2 * mid_point[1] - A[1]) return D # Example vertices A = (2, 4) B = (6, 4) C = (2, 8) print(find_fourth_vertex(A, B, C))

The output is:

(6, 8)

In the function `find_fourth_vertex()`

, we calculate the midpoint of the diagonal from B to C, then use vector subtraction to get the vector A to midpoint, which we then double and subtract from A to find D. This method is both elegant and efficient, leveraging mathematical properties of rectangles.

## Method 2: Slope Comparison

Since the opposite sides of a rectangle are parallel, we can use the slope between points to find the fourth vertex. By checking the slopes formed by the provided coordinates, we can deduce the directions in which to find the fourth vertex. We define a function `find_vertex_with_slopes()`

that uses the slope information to compute the fourth vertexβs location.

Here’s an example:

def find_vertex_with_slopes(A, B, C): if A[0] != B[0]: slope_AB = (A[1] - B[1]) / (A[0] - B[0]) if slope_AB == 0: # A, B, D are collinear D = (C[0], A[1]) else: D = (A[0], C[1]) else: D = (C[0], B[1]) return D # Example vertices print(find_vertex_with_slopes(A, B, C))

The output is:

(6, 8)

The `find_vertex_with_slopes()`

function examines the slopes of the sides formed by the given points to identify parallel sides and, consequently, the coordinates of the unknown vertex. This method is geometrically intuitive but may require additional checks to handle vertical lines, where the slope is undefined.

## Method 3: Using Complex Numbers

In this geometric approach, we model the vertices of the rectangle as complex numbers. We can use complex number arithmetic to rotate points around one another and find the fourth vertex. The function `find_vertex_with_complex()`

leverages this to calculate the missing vertex without directly dealing with slope calculations.

Here’s an example:

def find_vertex_with_complex(A, B, C): A, B, C = complex(*A), complex(*B), complex(*C) D = B + C - A return (D.real, D.imag) # Example vertices print(find_vertex_with_complex(A, B, C))

The output is:

(6.0, 8.0)

The `find_vertex_with_complex()`

function converts the points into complex numbers, simplifying the rotation and translation necessary to find the fourth vertex. This method provides a clear and concise code. However, an understanding of complex numbers is required to grasp the logic behind the operations.

## Method 4: Perpendicularity and Distance

If we understand that the diagonals of a rectangle are both equal in length and bisect each other, we can find the fourth vertex by ensuring the sides formed with the new point are perpendicular to existing sides and have proper lengths. A function `find_vertex_perpendicular()`

can be implemented to iterate over possible coordinates satisfying these conditions.

Here’s an example:

def find_vertex_perpendicular(A, B, C): # Calculate vectors AB = (B[0] - A[0], B[1] - A[1]) AC = (C[0] - A[0], C[1] - A[1]) # Find potential D by assuming it's perpendicular to B D = (A[0] + AC[0], A[1] + AB[1]) return D # Example vertices print(find_vertex_perpendicular(A, B, C))

The output is:

(6, 8)

In the `find_vertex_perpendicular()`

function, we exploit the property that diagonals are equal in length and bisect each other. Here, we assume point D will be perpendicular to B regarding A, which aids in calculating the coordinates. This method could, however, be more computationally intensive due to its iterative nature.

## Bonus One-Liner Method 5: Set Theory with Tuples

This concise method utilizes set theory and tuple manipulation. By creating sets of potential x and y coordinates based on the given vertices and finding the symmetrical differences, we immediately arrive at the missing vertex. It’s an efficient one-liner solution in Python, assuming we know the two vertices that share an edge with the target vertex.

Here’s an example:

D = (set([A[0], B[0], C[0]]) ^ set([A[0], C[0]]), set([A[1], B[1], C[1]]) ^ set([A[1], B[1]])) print(D)

The output is:

({6}, {8})

This elegant one-liner uses set theory to find the missing coordinate. We use the symmetric difference operator `^`

to find the elements that are in one set or the other but not in both, effectively giving us the missing x and y coordinates. This method is very succinct but assumes a certain level of familiarity with set theory and may require additional steps to format the output as a tuple.

## Summary/Discussion

**Method 1:**Vector Addition and Subtraction. Utilizes well-known geometrical properties; efficient and concise. May not be the most intuitive for those unfamiliar with vector operations.**Method 2:**Slope Comparison. Relies on the concept of parallel sides in a rectangle; can intuitively find the vertex. May require additional handling of undefined slopes for vertical lines.**Method 3:**Using Complex Numbers. Simplifies rotation and translation to a matter of arithmetic operations. Requires understanding of complex numbers, which might not be common knowledge.**Method 4:**Perpendicularity and Distance. Ensures side lengths and angles fit a rectangle’s properties. Possibly less efficient due to iteration over coordinates.**Method 5:**Set Theory with Tuples. Offers a very concise code; exploits Python’s powerful set operations. Assumes understanding of set theory and might need formatting for output.