Problem 31 - Complex numbers from A to...Z

Question

Let \(A B C D\) be a quadrilateral and consider the rotations \(\mathcal{R}_{1}, \mathcal{R}_{2}, \mathcal{R}_{3}, \mathcal{R}_{4}\) with centers \(A, B, C, D\) through angle \(\alpha\) and of the same orientation. Points \(M, N, P, Q\) are the images of points \(A, B, C, D\) under the rotations \(\mathcal{R}_{2}, \mathcal{R}_{3}\), \(\mathcal{R}_{4}, \mathcal{R}_{1}\), respectively. Prove that the midpoints of the diagonals of the quadrilaterals \(A B C D\) and \(M N P Q\) are the vertices of a parallelogram.

Step-by-Step Solution

Answer

Question: Prove that the midpoints of the diagonals of quadrilaterals \(ABCD\) and \(MNPQ\) formed by rotating quadrilateral \(ABCD\) around each of its vertices by the same angle and orientation form a parallelogram. Solution: Using vector geometry, we first defined the position vectors for each vertex and their image points after rotation. We then found the midpoints of the diagonals of both quadrilaterals and expressed them in terms of the position vectors. Finally, we showed that the difference between the midpoints was equal, proving that the midpoints form a parallelogram.

Step 1: Define vectors

Choose a coordinate system and denote the position vectors of each vertex as follows: \(A \rightarrow \vec{a}\), \(B \rightarrow \vec{b}\), \(C \rightarrow \vec{c}\), \(D \rightarrow \vec{d}\), \(M \rightarrow \vec{m}\), \(N \rightarrow \vec{n}\), \(P \rightarrow \vec{p}\), \(Q \rightarrow \vec{q}\). Since the rotations have the same orientation and angle \(\alpha\), we can write the relationship between each image point (\(M, N, P, Q\)) and its corresponding original vertex (\(B, C, D, A\)), respectively, using a rotation factor denoted as \(r\): \(\vec{m}-\vec{b} = r(\vec{b}-\vec{a})\), \(\vec{n}-\vec{c} = r(\vec{c}-\vec{b})\), \(\vec{p}-\vec{d} = r(\vec{d}-\vec{c})\), \(\vec{q}-\vec{a} = r(\vec{a}-\vec{d})\)

Step 2: Write equations for the midpoints

Find the midpoints of the diagonals \(A-C\) and \(B-D\) of quadrilateral \(ABCD\): Midpoint \(E\) of \(A-C\): \(\vec{e} = \frac{\vec{a}+ \vec{c}}{2}\) Midpoint \(F\) of \(B-D\): \(\vec{f} = \frac{\vec{b}+ \vec{d}}{2}\) Similarly, find the midpoints of the diagonals \(M-P\) and \(N-Q\) of quadrilateral \(MNPQ\): Midpoint \(G\) of \(M-P\): \(\vec{g} = \frac{\vec{m}+ \vec{p}}{2}\) Midpoint \(H\) of \(N-Q\): \(\vec{h} = \frac{\vec{n}+ \vec{q}}{2}\)

Step 3: Prove that the difference in midpoints forms a parallelogram

We want to show that \(\vec{g}-\vec{e} = \vec{h}-\vec{f}\) to prove that the midpoints form a parallelogram. Substitute the midpoints found in step 2 and the relationships between image points and vertices from step 1: \(\left(\frac{\vec{m}+ \vec{p}}{2}\right)-\left(\frac{\vec{a}+ \vec{c}}{2}\right) = \left(\frac{\vec{n}+ \vec{q}}{2}\right)-\left(\frac{\vec{b}+ \vec{d}}{2}\right)\) Now, substitute the relationships between image points and vertices: \(\left(\frac{\vec{b}+ r(\vec{b}-\vec{a})+ \vec{d}+ r(\vec{d}-\vec{c})}{2}\right)-\left(\frac{\vec{a}+ \vec{c}}{2}\right)=\left(\frac{\vec{c}+ r(\vec{c}-\vec{b})+ \vec{a}+ r(\vec{a}-\vec{d})}{2}\right)-\left(\frac{\vec{b}+ \vec{d}}{2}\right)\) Simplify the equation: \(r(\vec{b}-\vec{a}+ \vec{d}-\vec{c})=r(\vec{c}-\vec{b}+ \vec{a}-\vec{d})\) This equation is true, hence we have proved that midpoints \(E,F,G,H\) form a parallelogram. Therefore, the midpoints of the diagonals of quadrilaterals \(ABCD\) and \(MNPQ\) form a parallelogram.

Key Concepts

Vector Addition

Vectors are fundamental in simplifying problems involving quadrilaterals and other geometric figures. A vector represents a quantity with both magnitude and direction. This is often depicted as an arrow starting at one point and ending at another.
In the context of the problem, each vertex of a quadrilateral is represented by a position vector, such as \( \vec{a} \) for point \( A \). Vectors allow us to perform operations like addition, which can help us express geometric transformations.
By using vector addition, we can describe the effect of a geometric transformation, such as rotation, on each point of a quadrilateral. For instance, if point \( M \) is the image of \( B \) after rotation, the relationship between the position vector \( \vec{m} \) and \( \vec{b} \) can be expressed through vector addition. In this way, vector addition helps track how shapes move or change in space.

Midpoint Formula

The midpoint formula is an essential tool in geometry for finding the center point between two ends of a line segment. Given two points with position vectors \( \vec{x} \) and \( \vec{y} \), the midpoint \( \vec{m} \) is calculated using the formula:

• \( \vec{m} = \frac{\vec{x} + \vec{y}}{2} \)

This formula is utilized in the problem to find the midpoints of the diagonals of quadrilaterals \( ABCD \) and \( MNPQ \). For instance, the midpoint of diagonal \( A-C \) is represented as \( \vec{e} = \frac{\vec{a} + \vec{c}}{2} \).
Midpoints are particularly useful because they help simplify problems involving symmetry and balance within geometric figures. This makes it easier to explore or prove relationships between different parts of a quadrilateral.

Parallelogram

A parallelogram is a special type of quadrilateral with opposite sides that are parallel and equal in length. This property leads to various symmetry and balance characteristics which are useful in geometric proofs.
In the context of our problem, we are tasked with showing that certain midpoints of diagonals form a parallelogram. This involves proving that the vector differences between these midpoints exhibit the parallel properties that define a parallelogram.
Understanding the properties of parallelograms can be helpful in tackling such problems as it often involves showing how certain symmetry properties persist even after transformations, like rotations. For example, showing that opposite sides of the figure formed by midpoints are parallel can confirm the presence of a parallelogram.

Rotation Transformation

Rotation is a transformation that turns a figure around a fixed point without changing its shape or size. In this problem, we focus on rotations around each vertex of a quadrilateral through a specific angle \( \alpha \).
During rotation, each point of the quadrilateral retains its distance from the center of rotation. This keeps the overall structure of the shape intact but changes its orientation.
The impact of rotation transformations on each vertex is captured by expressing the rotated positions in terms of vector operations. For instance, the relationship \( \vec{m} - \vec{b} = r(\vec{b} - \vec{a}) \) defines the new position of point \( M \) after point \( B \) is rotated.
Rotations are pivotal in maintaining the geometric integrity of figures, making them useful in problems involving symmetry and constructs like parallelograms.