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We present results from a study of the distances and distribution of a sample of intermediate-age clusters in the Large Magellanic Cloud. Using deep near-infrared photometry obtained with ISPI on the CTIO 4m, we have measured the apparent K-band magnitude of the core helium burning red clump stars in 17 LMC clusters. We combine cluster ages and metallicities with the work of Grocholski & Sarajedini to predict each cluster's absolute K-band red clump magnitude, and thereby calculate absolute cluster distances. An analysis of these data shows that the cluster distribution is in good agreement with the thick, inclined disk geometry of the LMC, as defined by its field stars. We also find that the old globular clusters follow the same distribution, suggesting that the LMC's disk formed at about the same time as the globular clusters, ~ 13 Gyr ago. Finally, we have used our cluster distances in conjunction with the disk geometry to calculate the distance to the LMC center, for which we find (m-M)o = 18.40 +/- 0.04_{ran} +/- 0.08_{sys}, or Do = 47.9 +/- 0.9 +/- 1.8 kpc.