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A reach weapon is a melee weapon that allows its wielder to strike at targets that aren't adjacent to him. Most reach weapons double the wielder's natural reach, meaning that a typical Small or Medium wielder of such a weapon can attack a creature 10 feet away, but not a creature in an adjacent square. A typical Large character wielding a reach weapon of the appropriate size can attack a creature 15 or 20 feet away, but not adjacent creatures or creatures up to 10 feet away.
The horse and rider are both adjacent to you, so you cannot strike them at all.
Even if you have a clear line to its far square? If you are diagonally adjacent to the horse's head. It's back leg is 10' away.A reach weapon is a melee weapon that allows its wielder to strike at targets that aren't adjacent to him. Most reach weapons double the wielder's natural reach, meaning that a typical Small or Medium wielder of such a weapon can attack a creature 10 feet away, but not a creature in an adjacent square. A typical Large character wielding a reach weapon of the appropriate size can attack a creature 15 or 20 feet away, but not adjacent creatures or creatures up to 10 feet away.The horse and rider are both adjacent to you, so you cannot strike them at all.
Would doing so (assuming it's a legal action) grant the horse soft cover against your attack?
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If you want to attack the square itself (as in, the ground), you're perfectly capable.
But the rules for reach weapons do not divvy up creatures into their component squares.
The horse and rider are adjacent to you, thus they cannot be attacked with a reach weapon.
You can, however, "shorten your grip", and attack with the handle of your reach weapon, instead (as an improvised weapon).
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I'd like to weigh in on this. My take is a bit different. Please correct me if I'm wrong.
C may not attack the Horse, H, with a reach weapon, because the creatures are adjacent.
C may, however, attack the RIDER of the Horse, not pictured, either with a reach weapon or with an adjacent weapon. There may be soft cover, from the mount. My understanding is that a mounted rider shares the spaces of the mount. When the rider must be localized to a square, the rider's square is chosen by whomever is attacking. If the Rider is attacking then the Rider gets to choose from which square to attack. If Creature C is attacking the Rider, Creature C gets to decide which of the mount's squares to attack. This means that, for example, anything able to get adjacent to the mount can also attack the rider as if adjacent. The rider can not say, "I'm in this other square of the mount you can't reach, so nyaa!".
The Rider of the Horse could attack Creature C with either an adjacent weapon or with a reach weapon, just by choosing an appropriate square from which to attack. In some geometrical arrangements the Horse would provide soft cover. Soft cover normally negates AoOs, unless mount and rider both have Paired Opportunists in which case AoOs still happen but at -4 for soft cover.
I'm not certain the above is correct. Can someone please help find the relevant rules quotes? Hint: I'm looking for the basic rules about which square of the mount's footprint the rider occupies when. There are no rules clarifications re. the Paired Opportunists bit, and none are needed.
P.s. Make a link by typing this, only without the spaces:
[ u r l = "http://www.d20pfsrd.com/feats/combat-feats/paired-opportunists-combat-teamwork" ]Paired Opportunists[ / u r l ]
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A rider shares all squares with their mount.
A rider cannot, as you described it, pick its most advantageous square to attack from.
If they could, many Mounted Combat builds would rejoice, as it would alleviate the problems encountered when attacking at 10ft reach while your Horse/Bison/Tiger/Wolf mount only has 5ft reach.