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Step 2: {Ox.f, Oy.f, d.f}
{Ox, Oy, d}.
Step 3: {P = INTERSECT(Ox,d), d, Ox}
{P = [4, 0]}.
Step 4: {d, Oy, Q = INTERSECT(Oy,d)}
{Q = [0, 3]}.
Step 5: {P = [4, 0], Q = [0, 3]}
{P, Q}.
Step 6: {P, Q}
{MIDPOINT(P,Q) = [2, 3/2]}.
Step 7: {d, H = PROJECTION(O,d), O}
{ H = [36/25, 48/25]}.
Step 8: {H = [36/25, 48/25]}
{H}.
Example 5.2
: Given two points P(2, 5) and Q(5,1). Suppose d is a line that contains the point
P, and the distance between Q and d is 3. Find the equation of line d.
Specification of the problem:
Objects = {[P, point], [Q, point], [d, line]}.
Hypothesis = {DISTANCE(Q, d) = 3, P = [2, 5], Q = [5, 1], ["BELONG", P, d]}.
Goal = [d.f ].
Solution found by the system:
Step 1: {P = [2, 5]}
{P}.
Step 2: {DISTANCE(Q, d) = 3}
{DISTANCE(Q, d)}.
Step 3: {d, P}
{2d[1]+5d[2]+d[3] = 0}.
Step 4: {DISTANCE(Q, d) = 3}

5d[1]+d[2]+d[3]
3
.
2
2
d[1]
d
[2]

5d[1]+d[2]+d[3]
d[1]=1, 2d[1]+5d[2]+d[3]=0,
3
Step 5:
2
2
d[1]
d
[2]
24
134
d.f=(x+
y
=0),d.f=(x2=0)
7
7
24
134
Step 6:
d.f=(x+
y
=0),d.f=(x2=0)
7
7
{ d.f }
Example 5.3
: 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.
Specification of the problem:
Objects = {[A, POINT], [B, POINT], [C, POINT], [D, POINT], [M, POINT], [N, POINT],
[O1, PARALLELOGRAM[A, B, C, D], [O2, TRIANGLE[A, B, M]],
[O3, TRIANGLE [C, D, N]]}.
Hypothesis = { [« BELONG », M, SEGMENT[A, C]],