Proof Concerning Central Line \(X_5X_6\) of Triangle

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Hello everyone. Today we will prove theorem about central line \(X_5X_6\) of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let \(ABCD\) be a golden rectangle where \( \frac{AB}{BC} = \phi \), and construct the square \( BCQP \) inside it. Reflect \( P \) over \( D \) to obtain \( E \). Then, the line \( EB \) coincides with the central line \( X_5X_6 \) of triangle \( ABP \). Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: \( A = (0,0) \), \( B = (\phi x, 0) \), \( C = (\phi x, x) \), \( D = (0, x) \). The square \( BCPQ \) ensures \( P = (\phi x, 2x) \). The reflection of \( P \) across \( D \) is \( E = (-\phi x, 2x) \). 2. Compute the Nine-Point Center \( X_5 \) The nine-point center \( X_5 \) is the circumcenter of the medial triangle, which consists of the midpoints: \[ M_1 = \left(\frac{0 + \phi x}{2}, 0\right) = \left(\frac{\phi x}{2}, 0\right), \] \[ M_2 = \left(\f...
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Proof that a natural number can be expressed as a product of prime numbers

Hello everyone. Today we will prove that any natural number greater than \(1 \) can be expressed as a product of a prime number and number one or as a product of few prime numbers. We will use the method of mathematical induction as a proof method. 

Theorem: Let \(n \) be a natural number greater than \(1 \). Then \(n \) can be expressed as the product of one prime number and number one or as the product of few prime numbers.

Proof:

Note that if \(n \) is a prime number, the statement is automatically proved because any number can be written as the product of that number and number one. 

1. Base case (n = 2) 

Since \(2 \) is a prime number the statement is automatically proved. 

2. Induction hypothesis (n = m) 

Suppose that \(\forall k \in \mathbb {N}, \quad 2 \le k \le m \), \(k \) can be expressed as the product of a prime number and number one or as a product of few prime numbers. 

3. Inductive step (n = m + 1) 

Using the assumption from the second step, we will prove that: \(\forall k \in \mathbb {N}, \quad 2 \le k \le m + 1 \), \( k \) can be expressed as the product of one prime number and number one or as the product of few prime numbers. 

For all \(k \) less than \(m + 1 \) the truth of the statement automatically follows from the inductive hypothesis. Let us now consider the case when \(k = m + 1 \). If \(m + 1 \) is a prime number, the statement is automatically proved. Otherwise \(m + 1 \) is a composite number which can be expressed in the form \(m + 1 = pq \), where \(p \) and \(q \) are natural numbers such that  \(2 \le p <m+1\) and \(2 \le q <m+1\) , i.e. \(2 \le p \le m\) and \(2 \le q \le m\) . According to the inductive hypothesis, both numbers \(p \) and \(q \) can be expressed as products of few primes or as products of one prime number and number one, therefore the same is true for the number \(m + 1 \). 

\(\blacksquare \)

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