The term "altitude" is the same as "height of a triangle". It is perpendicular to the base. Since we can rotate the triangle to have any side be horizontal, there are effectively 3 possible bases. Hence, there are 3 heights. It all depends how you look at it.
Let h1, h2, and h3 be the three altitudes or heights.
Without loss of generality, we'll focus on the first two heights h1 and h2. Their respective bases are b1 and b2.
If we use b1 as the base, then the area is...
area = 0.5*base*height = 0.5*b1*h1
Similarly, the other base gives the area of:
area = 0.5*b2*h2
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Since both formulas refer to the same area (because we're working with the same triangle), we can set the expressions equal to one another
0.5*b1*h1 = 0.5*b2*h2
b1*h1 = b2*h2
Let's see what happens when b1 = b2, so,
b1*h1 = b2*h2
b1*h1 = b1*h2
b1h1 - b1h2 = 0
b1(h1 - h2) = 0
b1 = 0 or h1 - h2 = 0
b1 = 0 or h1 = h2
If the bases b1 and b2 were equal, then either those bases must be 0 which isn't possible, or the altitudes must be equal. However, the initial premise is that the heights must be different from one another.
Therefore, the bases b1 and b2 can't be the same length.
We could follow the same steps and logic to conclude that if the altitudes h1 and h3 were different, then the bases b1 and b3 can't be the same. Similarly, we would conclude that b2 and b3 can't be the same. This is where the "without loss of generality" kicks in.
In other words, we only need to focus on one subcase to extend the logic to the other cases, without having to actually do every single step. That would be a bit tedious busywork.
In conclusion, we've shown that if the heights are different, then their respective bases must be different. This leads to wrapping up the proof that we have a scalene triangle.
Side note: I used an indirect proof or proof by contradiction. I assumed that a non-scalene triangle was possible and it led to a contradiction of h1 = h2.
