Continuing to work with the set of $(x,y)$ points from this lesson, let's look for some triangles amongst the points.

On the one hand, it's kind of easy to find triangles, because any 3 points will form a triangle. But, we still want Prolog to do it, so we'll get a bit of practice with it. The goal goal: line([P1,P2]), line([P2,P3]), line([P3,P1]). will find triangles, noting that we ask Prolog to look for 3 lines that are formed solely of 3 unique points ($P_1$, $P_2$ and $P_3$). Notice we did not do line([P1,P2]), line([P3,P4]), line([P5,P6]).

To make Prolog work a little harder, suppose we restricted our triangle search to right triangles? Right triangles also need the three lines the goal above finds, with the additional qualification that two of the 3 lines are perpendicular.

Take a look at the goal below.

point(-4,4).

point(-3,2).

point(-1,1).

point(0,0).

point(-3,0).

point(1,3).

point(3,3).

point(-4,-4).

point(3,-3).

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line([(X1,Y1),(X2,Y2)]) :- point(X1,Y1), point(X2,Y2), [X1,Y1] \= [X2,Y2].

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horiz([(_,Y),(_,Y)]).

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vert([(X,_),(X,_)]).

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is_neg1(X) :- D is abs(X+1), D < 0.1.

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perp(L1,L2) :- horiz(L1), vert(L2).

perp(L1,L2) :- vert(L1), horiz(L2).

perp(L1,L2) :- \+ vert(L1), \+ vert(L2), slope(L1,M1), slope(L2,M2), P is M1*M2, is_neg1(P).

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slope([(X,_),(X,_)],infinite) :- !.

slope([(X1,Y1),(X2,Y2)],M) :- M is (Y2-Y1)/(X2-X1).

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goal: line([P1,P2]), line([P2,P3]), line([P3,P1]), perp([P1,P2],[P2,P3]).