In a symmetric distribution, are the mean, median, and mode necessarily equal?
Yes, they are equal to one another.
In a symmetric distribution, one half of the curve is the mirror image of the other half.
The frequencies are distributed evenly. The pull is the same on both sides.
Since one side balances the other, all the three measures fall exactly at the middle.
Symmetry (in Geometry) means this. From whatever angle or the direction, you look at a shape, it presents the same view point. For example, when you look at a square, there are two ways of looking at it. length-wise or breadth-wise. Since the length and breadth are equal, the view does not differ and hence it is symmetric.
In a similar way, when we study a probability distribution, we may study it from either of the measures of central tendency - mean, median, or mode. Unless these values are identical, we cannot say that the distribution is symmetrical.