/*----------------------------------------------------------------
	Basic function for construction of equations 
	in Geometric Theorem proving 
----------------------------------------------------------------*/
put('geometry2,1,'version);

generate_ideal(hypotheses,conclusions):=
	append(hypotheses,
		makelist(1-genpar()*conclusions[i],i,1,length(conclusions)));

/* X, Y, Z on the plane are colinear */
collinear(X,Y,Z):= det(matrix(cons(1,X),cons(1,Y),cons(1,Z)));

/* Vector AB and CD are perpendicular */
perpendicular(A,B,C,D):= (B-A).(D-C);

/* AB and CD have equal length */
equidistant(A,B,C,D):=(B-A).(B-A)-(D-C).(D-C);

/* AB and CD are parallel */
parallel(A,B,C,D):=det(matrix(B-A,D-C));

/* M is the midpoint of AB; 2 equations */
midpoint(M,A,B):=2*M-(A+B);

/* Points A and B are identical */
identical(A,B):=B-A;

/* D is the distance between A and B */
distance(A,B,D):=D^2-(B-A).(B-A);

/* Angle P with cos COS_P and sin SIN_P is the angle ABC
where segments AB and BC have lengths D_AB and D_BC */
angle(COS_P,SIN_P,A,B,C,D_AB,D_BC):=
	[
		cos_p^2+sin_p^2-1,
		distance(A,B,D_AB),
		distance(B,C,D_BC),
		COS_P*D_AB*D_BC-(C-B).(A-B),
		SIN_P*D_AB*D_BC-det(matrix(C-B,A-B))
	];

/* Point X is on ellipse with center at O with semi-axes a and b */
ellipse(X,O,a,b):= b^2*(X[1]-O[1])^2+a^2*(x[2]-o[2])^2-a^2*b^2;