Geometric Problems: Can You Help Me Find Solutions?

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In summary, the conversation discusses several geometry problems, including proving that all triangles formed by sides an, bn, and cn are isosceles, finding the sides of a triangle circumscribed by a circle with a radius of 6.25, proving that the center of an inscribed circle lies on a line that splits a triangle into two figures with equal perimeters and areas, and proving that the area of an octagon formed by connecting the vertices of a parallelogram with the centers of the two opposite sides is one sixth the area of the parallelogram. The conversation also includes hints and suggestions for solving the problems.
  • #1
Ebn_Alnafees
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Problem 1
Prove that if for {a, b, c} R and all n N there exists
a triangle with the sides an, bn and cn, then all of these triangles
are isoscoles.


Problem 2
A circle with a radius of 6.25 is circumscribed around a triangle
with the sides a, b and c. Find these sides, if {a, b, c} N.

Problem 3
A line splits a triangle into two new figures with equal perimeters
and areas. Prove that the center of the inscribed circle lies on this line.


Problem 4
The eight lines that connect the vertices of a parallelogram with the
centers of the two opposite sides form an octogon. Prove that the
octogon's area is exactly one sixth the area of the parallelogram



please guys I'm a new member here can you help me in these geometry problems?
 
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  • #2
1] Why should it be isosceles?
Any three sides form a triangle if,
sum of any two sides is greater than the third side

Once such a triangle is formed i guess any integer multiple of the lengths work!
a+b<c
then na+nb<nc

2] Probably some clumsy work in the offing here,
a/sinA = b/sinB = c/sinC = 2R
find cosines using the cosine rule, convert them to sines and sub it above.
So you should find 3 bizarre equations in 3 unknowns, now the biggest trouble is solving them for integer solutions?? I am not sure if that's any easier :uhh:

-- AI
P.S-> The 3rd question seems interesting, must check it
 
  • #3
thanx tenaliraman
 
  • #4
any 1 for 3 and 4
 
  • #5
problem 3
works only with equilateral triangles
 
  • #6
come to think of it. doesn't work at all.
 
  • #7
Ebn_Alnafees, you should give the problems a try and post as much of your work as possible here. Read the sticky thread.
Problem 3: Hint : Draw one bisector that cut the line. Call the intersection of that bisector and the line E. What do you reckon? If the line cuts 2 legs AB (at F), and AC (at G), then the bisector should go through A, so the distance between E and AB, AC is the same.
Use what the problem tells you : The two new figure have the same perimeters and area.
Hope you get it.
Viet Dao,
 
Last edited:
  • #8
can you email me the answer. from how i understand the question. there's no solution. except the special cases of isosolese and equilateral triangles split congruently.
 

1. How do you prove that if a number is divisible by 3, it is also divisible by 9?

To prove this, we can use the transitive property of division. If a number, let's say x, is divisible by 3, then we can write it as x = 3y, where y is some other integer. Similarly, if x is divisible by 9, we can write it as x = 9z, where z is another integer. Now, substituting the first equation into the second, we get 3y = 9z, which can be simplified to y = 3z. This shows that z is also an integer, therefore, if x is divisible by both 3 and 9, it must also be divisible by 9.

2. Can you prove that if a and b are both even numbers, their sum is also even?

Yes, we can prove this using the definition of even numbers. An even number is any integer that can be divided by 2 without leaving a remainder. So, we can express a and b as a = 2x and b = 2y, where x and y are integers. Now, their sum, a + b = 2x + 2y = 2(x + y), which is also divisible by 2 without leaving a remainder. Therefore, the sum of two even numbers is always even.

3. How do you prove that if a and b are both odd numbers, their product is an odd number?

We can prove this by contradiction. Suppose a and b are both odd numbers and their product, ab, is an even number. This means that ab can be written as ab = 2k, where k is an integer. Now, since a is an odd number, we can write it as a = 2m + 1, where m is an integer. Similarly, b = 2n + 1, where n is an integer. Substituting these values into the equation ab = 2k, we get (2m + 1)(2n + 1) = 2k, which simplifies to 4mn + 2m + 2n + 1 = 2k. This means that 2 divides (4mn + 2m + 2n), which is not possible since 2 only divides even numbers. Therefore, our assumption was wrong and ab must be an odd number.

4. Is it possible to prove that if a number is a perfect square, its square root is also an integer?

Yes, this is known as the definition of a perfect square. A perfect square is any number that can be written as the product of two equal integers. So, if we take the square root of a perfect square, we will get the same integer as both the factors. For example, the perfect square 25 can be written as 5 x 5, so its square root is 5. This holds true for all perfect squares.

5. How can you prove that if a and b are both prime numbers, their sum is not a prime number?

We can prove this using a counterexample. Let's take a = 3 and b = 5, both of which are prime numbers. Their sum, 3 + 5 = 8, is not a prime number as it is divisible by 2. Therefore, the statement is not always true. However, it is true for all other possible combinations of two prime numbers.

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