Abstract: The decays B+→ J/ψπ+π−K+ are studied using a data set corresponding to an integrated luminosity of 9 fb−1 collected with the LHCb detector in proton-proton collisions between 2011 and 2018. Precise measurements of the ratios of branching fractions with the intermediate ψ2(3823), χc1(3872) and ψ(2S) states are reported. The values areBB+→ψ23823K+×Bψ23823→J/ψπ+π−BB+→χc13872K+×Bχc13872→J/ψπ+π−=3.56±0.67±0.11×10−2, BB+→ψ23823K+×Bψ23823→J/ψπ+π−BB+→ψ2SK+×Bψ2S→J/ψπ+π−=1.31±0.25±0.04×10−3, BB+→χc13872K+×Bχc13872→J/ψπ+π−BB+→ψ2SK+×Bψ2S→J/ψπ+π−=3.69±0.07±0.06×10−2, where the first uncertainty is statistical and the second is systematic. The decay of B+→ ψ2(3823)K+ with ψ2(3823) → J/ψπ+π− is observed for the first time with a significance of 5.1 standard deviations. The mass differences between the ψ2(3823), χc1(3872) and ψ(2S) states are measured to be mχc13872−mψ23823=47.50±0.53±0.13MeV/c2, mψ23823−mψ22S=137.98±0.53±0.14MeV/c2, mχc13872−mψ22S=185.49±0.06±0.03MeV/c2, resulting in the most precise determination of the χc1(3872) mass. The width of the ψ2(3823) state is found to be below 5.2 MeV at 90% confidence level. The Breit-Wigner width of the χc1(3872) state is measured to be Γχc13872BW=0.96−0.18+0.19±0.21MeV which is inconsistent with zero by 5.5 standard deviations.