Prove that there exist exactly two non-similar isosceles triangles `A B C` such that `tanA+tanB+tanC=100.`

Explanation

Let A=B
`2A+C=180^@`
2A=-tanC
Also we have tanA+tanB+tanC=100 ltbr. 2tanA+tanC=100
From (1) and (2) we have
2tan A -Tan 2A=100
`rArr 2tan A-(2tanA)/(1-tan A)-100=0`
Let tan A=x
So we have `x^(3)-50x^(2)+50=0`
Let `f(x)=x^(3)-50x^(2)+50`
`f(x)=3x^(2)-100x=0`
`x=0,(100)/(3)`
Also f(x)=50
So graph of function is as shown in the following figure

Clearly f(x) =0 has exactly three distinct real roots
Therefore ,tan A and hence A has threee distinct values but one of them will be obtuse anlgle as one of the roots of the equation f(x)=0 negative.
Hence there exist exactly two non similar isiceles triangles.