MHB Prove that the radius of the incircle of △ is rational

[lfdahl]

Let $\bigtriangleup$ be an isosceles triangle for which the length of a side and the length of the base are rational. Prove that the radius of the incircle of $\bigtriangleup $ is rational if and only if the two right triangles formed by the altitude to the base are similar to a right triangle with integer side lengths

 

[Albert1]

Let $\bigtriangleup$ be an isosceles triangle for which the length of a side and the length of the base are rational. Prove that the radius of the incircle of $\bigtriangleup $ is rational if and only if the two right triangles formed by the altitude to the base are similar to a right triangle with integer side lengths

my solution:

Spoiler

given isosceles triangle $ABC$
$h^2=x^2-\dfrac {y^2}{4}$
$h=\sqrt{\dfrac {4x^2-y^2}{4}}$
$h$ must be a perfect square
$r=\dfrac {yh}{2x+y}$
for $x,y,r$ being rational ,if x,y and h are all integers then we are done
else we can enlarge the original triangle to make them (x,y,h) all integers
so the two right triangles formed by the altitude to the base are similar to a right triangle (the bigger one) with integer side lengths
View attachment 7251

 

[lfdahl]

my solution:

Spoiler

given isosceles triangle $ABC$
$h^2=x^2-\dfrac {y^2}{4}$
$h=\sqrt{\dfrac {4x^2-y^2}{4}}$
$h$ must be a perfect square
$r=\dfrac {yh}{2x+y}$
for $x,y,r$ being rational ,if x,y and h are all integers then we are done
else we can enlarge the original triangle to make them (x,y,h) all integers
so the two right triangles formed by the altitude to the base are similar to a right triangle (the bigger one) with integer side lengths

Well done, Albert!Thankyou for your participation.