Proof Concerning Central Line X5X6X_5X_6 of Triangle

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Hello everyone. Today we will prove theorem about central line X5X6X_5X_6 of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let ABCDABCD be a golden rectangle where ABBC=ϕ \frac{AB}{BC} = \phi , and construct the square BCQP BCQP inside it. Reflect P P over D D to obtain E E . Then, the line EB EB coincides with the central line X5X6 X_5X_6 of triangle ABP ABP . Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: A=(0,0) A = (0,0) , B=(ϕx,0) B = (\phi x, 0) , C=(ϕx,x) C = (\phi x, x) , D=(0,x) D = (0, x) . The square BCPQ BCPQ ensures P=(ϕx,2x) P = (\phi x, 2x) . The reflection of P P across D D is E=(−ϕx,2x) E = (-\phi x, 2x) . 2. Compute the Nine-Point Center X5 X_5 The nine-point center X5 X_5 is the circumcenter of the medial triangle, which consists of the midpoints: M1=(0+ϕx2,0)=(ϕx2,0), 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 11 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 nn be a natural number greater than 11 . Then nn 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 nn 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 22 is a prime number the statement is automatically proved. 

2. Induction hypothesis (n = m) 

Suppose that ∀k∈N,2≤k≤m\forall k \in \mathbb {N}, \quad 2 \le k \le m , kk 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: ∀k∈N,2≤k≤m+1\forall k \in \mathbb {N}, \quad 2 \le k \le m + 1 , k k can be expressed as the product of one prime number and number one or as the product of few prime numbers. 

For all kk less than m+1m + 1 the truth of the statement automatically follows from the inductive hypothesis. Let us now consider the case when k=m+1k = m + 1 . If m+1m + 1 is a prime number, the statement is automatically proved. Otherwise m+1m + 1 is a composite number which can be expressed in the form m+1=pqm + 1 = pq , where pp and qq are natural numbers such that  2≤p<m+12 \le p <m+1 and 2≤q<m+12 \le q <m+1 , i.e. 2≤p≤m2 \le p \le m and 2≤q≤m2 \le q \le m . According to the inductive hypothesis, both numbers pp and qq 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+1m + 1 . 

■\blacksquare

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