Re:[EMHL] Perimeter things!
- Dear TigerSandre,
>1)Prove that two triangles are alike if and only if the perimeters,Let be s1=ks,R1=kR,r1=kr.
>the radius of the incircles and the radius of the circumcircles are
>proportional. Show that two triangles incriebed in the same circle
>which have the same perimeters and areas are congruent.
Sides of the first triangle are the roots of equation
Sides of the second triangle are the roots of equation
If we put x-->a in (1) and x-->a1 in (2) and take
(-k^3)(1)+(2) then we have
or (a1-ka)[(a1)^2+ka1(a-2s)+(k^2)bc]=0 or
Analogously for sides b,b1 and c,c1.
Thus we have that triangles are similar.
The second statement is the special case of the first part when k=1.
Maybe the easier proof is known?
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