In 1989 the basis transferred to you and your mom and your brother. Each of you had a basis of $3,333.33.
In 2013 when your mom died her third of the value stepped up to one third of the market value that year. So her basis became $6,666.66. That value transferred to you and your brother so each of you had a basis of $6,666.66.
In 2020 the whole basis of $13,333.33 transferred to your brother with your quitclaim.
In 2022 your brother quitclaimed it back to you and your ex-sister-in-law so the basis transferred back at $6,666.66 each.
In 2023 your brother passed but because he had quitclaimed it back to you before that his death has no effect on the basis.
In 2023 you sold the home for $155,000. That is $77,500 apiece, minus the $6,666.66 basis for a gain of $70,834.34 each.
There were also closing costs on the sale. As well as any money you spent fixing the place up in order to sell it. Those costs should be divided between you and your ex-sister-in-law and deducted from the gain.
As a purchase date you can use the date in 2022 when your brother quitclaimed the house back to you.
@Lindysue
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