Does the sale of a minority share trigger a tax assessment?

[Baird King]

My wife and I purchased a property with a partner in a 1/3, 1/3, 1/3 split.  We recently purchased his 1/3 share and I am wondering if that will trigger a tax assessment? We had to refinance in order to pull out cash to purchase his share, and Zillow now shows the amount we paid for his share as the most recent sale price, so clearly it is public record.  I read somewhere that tax assessment only gets triggered when a >50% share transfers ownership, but I have not confirmed.  Even if a transfer of a 1/3 share triggered an assessment, how would the assessor know the ownership split in order to assess the new value of the property based off the sale price.  For example, if we had purchased the property initially for $900k ($300k per person), but we bought his share for $400k, how would the assessor's office use the sale price of his share to modify our taxes?  As far as they know we could have had a 1%, 1%, 98% split and he could have just given us a super deal... Thanks!