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Since the triangle ABC is a right triangle, then one of it's angles is 90 degrees.
We'll put A = 90 degrees.
That means that B+C = 90 degrees.
We know that sin 90 = sin A = 1.
We'll re-write the condition given by enunciation:
sin A/(x+1) = sin B/(x^2 + 1) = sin C/2x
We'll substitute sin A by 1:
1/(x+1) = sin B/(x^2 + 1) = sin C/2x
We'll take the 2nd and the 3rd ratios and we'll use the following property:
a/b = c/d
(a+c)/(b+d) = c/d
sin B/(x^2 + 1) = sin C/2x
(sin B + sin C)/(x^2 + 2x + 1) = sin C/2x
(sin B + sin C)/(x + 1)^2 = sin C/2x
1/(x+1) = sin C/2x
sin C = 2x/(x+1)
(sin B + sin C)/(x + 1)^2 = 1/(x+1)
(sin B + sin C)/(x + 1) = 1
sin B + sin C = x+1
From all the identities above, we'll get 2x = x+1.
2x - x = 1
x = 1
The right angle triangles that satisfy the given constraint have x = 1.
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