In a symmetric distribution, are the mean, median, and mode always equal?
No. Not always. You need some more properties of symmetric distribution to say
In a distribution median and mode always exists but mean is not always exists. Consider Cauchy distribution, the mean doesn't exists. Mode always exists but may not be unique i.e. we may get distributions which are not unimodal (i.e. multimodal).
So, the conclusion is if we have a symmetric distribution whose mean exists and the distribution is unimodal then we can say
#" Mean " = " Median " = " Mode "#
Also mean, median and mode are the point of symmetry.