Prove that if for {a, b, c} R

[Ebn_Alnafees]

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?

 

[TenaliRaman]

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

 

[Ebn_Alnafees]

thanx tenaliraman

 

[Ebn_Alnafees]

any 1 for 3 and 4

 

[bao_ho]

problem 3
works only with equilateral triangles

 

[bao_ho]

come to think of it. doesn't work at all.

 

[VietDao29]

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,

 

[bao_ho]

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.