  We demonstrate computer-assisted proofs of "Kariya's theorem," a theorem in
elementary geometry, with computer algebra. In the proof of geometry theorem
with computer algebra, vertices of geometric figures that are subjects for the
proof are expressed as variables. The variables are classified into two
classes: arbitrarily given points and the points defined from the former points
by constraints. We show proofs of Kariya's theorem with two formulations
according to two ways for giving the arbitrary points: one is called "vertex
formulation," and the other is called "incenter formulation," with two methods:
one is Gr\"obner basis computation, and the other is Wu's method. Furthermore,
we show computer-assisted proofs of the property that the point so-called
"Kariya point" is located on the hyperbola so-called "Feuerbach's hyperbola",
with two formulations and two methods.

Symbolic Computation;Commutative Algebra