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geometry: in the triangle ABC, suppose that the length of edge BC = 5; the plane (P) has
the equation 2x + 3y - z + 6 = 0, and the point M has the coordinate (1, 2, 3).
Fact of kind 4 : equality on objects or attributes of objects. This kind of facts is also
popular, and there are many problems related to it on the knowledge base. The
following problem in plane geometry gives some examples for facts of kind 4.
Problem: Given the parallelogram ABCD. Suppose M and N are two points of segment AC
such that AM = CN. Prove that two triangles ABM and CDN are equal.
In the problem we have to determine equality between two C-objects, a fact of kind 4.
Fact of kind 5 : a dependence of an object on other objects by a general equation. An
example in geometry for this kind of fact is that w = 2*u + 3*v; here u, v and w are
vectors.
Fact of kind 6 : a relation on objects or attributes of the objects. In almost problems there
are facts of kind 6 such as the parallel of two lines, a line is perpendicular to a plane, a
point belongs to a line segment.
Fact of kind 7 : a determination of a function.
Fact of kind 8 : a determination of a function by a value or a constant expression.
Fact of kind 9 : equality between an object and a function.
Fact of kind 10 : equality between a function and another function.
Fact of kind 11 : a dependence of a function on other functions or other objects by an
equation.
The last five kinds of facts are related to knowledge about functions, the component Funcs
in the COKB model. The problem below gives some examples for facts related to functions.
Problem: Let d be the line with the equation 3x + 4y - 12 = 0. P and Q are intersection points
of d and the axes Ox, Oy.
a.
Find the central point of PQ
b.
Find the projection of O onto the line d.
For each line segment, there exists one and only one point which is the central point of that
segment. Therefore, there is a function MIDPOINT(A, B) that outputs the central point M of
the line segment AB. Part (a) of the above problem can be represented as to find the point I
such that I = MIDPOINT(P,Q), a fact of kind 9. The projection can also be represented by the
function PROJECTION(M, d) that outputs the projection point N of point M onto line d.
Part (b) of the above problem can be represented as to find the point A such that A =
PROJECTION(O,d), which is also a fact of kind 9.
Unification algorithms of facts were designed and used in different applications such as the
system that supports studying knowledge and solving analytic geometry problems, the
program for studying and solving problems in Plane Geometry, the knowledge system in
linear algebra.
3.1.4 Specification language for COKB model
The language for the COKB model is constructed to specify knowledge bases with
knowledge of the form COKB model. This language includes the following:
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