Molecules with ALMA at Planet-forming Scales (MAPS). IX. Distribution and Properties of the Large Organic Molecules HC$_3$N, CH$_3$CN, and $c$-C$_3$H$_2$

The precursors to larger, biologically-relevant molecules are detected throughout interstellar space, but determining the presence and properties of these molecules during planet formation requires observations of protoplanetary disks at high angular resolution and sensitivity. Here we present 0.3" observations of HC$_3$N, CH$_3$CN, and $c$-C$_3$H$_2$ in five protoplanetary disks observed as part of the Molecules with ALMA at Planet-forming Scales (MAPS) Large Program. We robustly detect all molecules in four of the disks (GM Aur, AS 209, HD 163296 and MWC 480) with tentative detections of $c$-C$_3$H$_2$ and CH$_3$CN in IM Lup. We observe a range of morphologies -- central peaks, single or double rings -- with no clear correlation in morphology between molecule nor disk. Emission is generally compact and on scales comparable with the millimetre dust continuum. We perform both disk-integrated and radially-resolved rotational diagram analysis to derive column densities and rotational temperatures. The latter reveals 5-10 times more column density in the inner 50-100 au of the disks when compared with the disk-integrated analysis. We demonstrate that CH$_3$CN originates from lower relative heights in the disks when compared with HC$_3$N, in some cases directly tracing the disk midplane. Finally, we find good agreement between the ratio of small to large nitriles in the outer disks and comets. Our results indicate that the protoplanetary disks studied here are host to significant reservoirs of large organic molecules, and that this planet- and comet-building material can be chemically similar to that in our own Solar System. This paper is part of the MAPS special issue of the Astrophysical Journal Supplement Series.