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Exactly! I have been saying that in few comments!
You don't need to have multiple CASE statements next to each other or nested CASEs. If you check the conditions in the right order, only one CASE statement with multiple WHENs will do the job:
SELECT CASE WHEN A + B <= C OR A + C <= B OR B + C <= A THEN 'Not A Triangle'
WHEN A = B AND B = C THEN 'Equilateral'
WHEN A = B OR A = C OR B = C THEN 'Isosceles'
I think this is the cleanest and shortest answer...
When the Isosceles and Equilateral case statements order are reversed, that is the Equi is checked after Isos then the test case fails. Is it a necessaity to do this in this ordeR?