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Two Similar Triangles


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Infinite number.

Note that 20,141 = 11x1831

Let the smaller triangle have a side 1831 and two equal other sides larger than 916, say 1000 and 1000.

Let the larger triangle have sides 12 times as large: 21972 and 12000 and 12000.

There are an infinite number of lengths for the equal sides other than 1000.

Therefore I assume that is not the correct interpretation of "two sides the same."

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Let triangle 1 have sides a b c where a <= b <= c, and triangle 2 have sides ar br cr.
Two pairs of sides are equal, so that the corresponding sides are


a b c
ar br cr


so r = b/a = c/b and the sides become:

a ar ar2
ar ar2 ar3


we have also ar3 - a = 20141 or r3 = (a + 2041)/a or

r = ((a + 20141)/a)1/3.

We need to solve for a and r.

More later.
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Let triangle 1 have sides a b c where a <= b <= c,

and triangle 2 have sides ar br cr.
Two pairs of sides are equal, so that the corresponding sides are

a b c
ar br cr


so r = b/a = c/b and the sides become:

a ar ar2
ar ar2ar3


we have also ar3 - a = 20141 or a = 20141/(r3 - 1) = integer.

r is rational, so we can write r = n/d where n and d are integers.

A simple search on integral values of n and d that gives integral values of a finds:

n = 30, d = 19, a = 6859, r = 1.578947368 ..., from which the triangle sides are.

6859 10830 17100

10830 17100 27000.

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