No. They are equal only if the distribution is symmetrical.
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Yes, and they WILL be if the distribution is symmetrical.
No, not always. Median is the number that's in the middle of a group of numerical data. Mean is just the average of a set of numbers, which isn't always in the middle.
Median.
If the triangle is really isosceles, and it's not lying on one of the equal sides, then the altitude is always a median.
All equal.