//
// A \
// \ B
// \ /
// O \/ X
// /\
// / \
// C / D
//
// Two tangents:
// Intersection(X, AD, BC), Tangent(Circle(O), BC), Tangent(Circle(O), AD) -> 2 * Angle(AXC) = MajorArc(AC) - MinorArc(AC)
//
public static List<EdgeAggregator> InstantiateTwoTangentsTheorem(Tangent tangent1, Tangent tangent2, Intersection inter, GroundedClause original1, GroundedClause original2)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
CircleSegmentIntersection tan1 = tangent1.intersection as CircleSegmentIntersection;
CircleSegmentIntersection tan2 = tangent2.intersection as CircleSegmentIntersection;
if (tan1.StructurallyEquals(tan2)) return newGrounded;
// Do the tangents apply to the same circle?
if (!tan1.theCircle.StructurallyEquals(tan2.theCircle)) return newGrounded;
Circle circle = tan1.theCircle;
// Do these tangents work with this intersection?
if (!inter.HasSegment(tan1.segment) || !inter.HasSegment(tan2.segment)) return newGrounded;
// Overkill? Do the tangents intersect at the same point as the intersection's intersect point?
if (!tan1.segment.FindIntersection(tan2.segment).StructurallyEquals(inter.intersect)) return newGrounded;
//
// Get the arcs
//
Arc minorArc = new MinorArc(circle, tan1.intersect, tan2.intersect);
Arc majorArc = new MajorArc(circle, tan1.intersect, tan2.intersect);
Angle theAngle = new Angle(tan1.intersect, inter.intersect, tan2.intersect);
//
// Construct the new relationship
//
NumericValue two = new NumericValue(2);
GeometricAngleArcEquation gaaeq = new GeometricAngleArcEquation(new Multiplication(two, theAngle), new Subtraction(majorArc, minorArc));
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(original1);
antecedent.Add(original2);
antecedent.Add(inter);
antecedent.Add(majorArc);
antecedent.Add(minorArc);
newGrounded.Add(new EdgeAggregator(antecedent, gaaeq, annotation));
return newGrounded;
}
//
// A \
// \ B
// \ /
// O \/ X
// /\
// / \
// C / D
//
// One Secant, One Tangent
// Intersection(X, AD, BC), Tangent(Circle(O), BC) -> 2 * Angle(AXC) = MajorArc(AC) - MinorArc(AC)
//
public static List<EdgeAggregator> InstantiateOneSecantOneTangentTheorem(Intersection inter, Tangent tangent, GroundedClause original)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
CircleSegmentIntersection tan = tangent.intersection as CircleSegmentIntersection;
// Is the tangent segment part of the intersection?
if (!inter.HasSegment(tan.segment)) return newGrounded;
// Acquire the chord that the intersection creates.
Segment secant = inter.OtherSegment(tan.segment);
Circle circle = tan.theCircle;
Segment chord = circle.ContainsChord(secant);
// Check if this segment never intersects the circle or doesn't create a chord.
if (chord == null) return newGrounded;
//
// Get the near / far points out of the chord
//
Point closeChordPt = null;
Point farChordPt = null;
if (Segment.Between(chord.Point1, chord.Point2, inter.intersect))
{
closeChordPt = chord.Point1;
farChordPt = chord.Point2;
}
else
{
closeChordPt = chord.Point2;
farChordPt = chord.Point1;
}
//
// Acquire the arcs
//
// Get the close arc first which we know exactly how it is constructed AND that it's a minor arc.
Arc closeArc = Arc.GetFigureMinorArc(circle, closeChordPt, tan.intersect);
// The far arc MAY be a major arc; if it is, the first candidate arc will contain the close arc.
Arc farArc = Arc.GetFigureMinorArc(circle, farChordPt, tan.intersect);
if (farArc.HasMinorSubArc(closeArc))
{
farArc = Arc.GetFigureMajorArc(circle, farChordPt, tan.intersect);
}
Angle theAngle = Angle.AcquireFigureAngle(new Angle(closeChordPt, inter.intersect, tan.intersect));
//
// Construct the new relationship
//
NumericValue two = new NumericValue(2);
GeometricAngleArcEquation gaaeq = new GeometricAngleArcEquation(new Multiplication(two, theAngle), new Subtraction(farArc, closeArc));
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(original);
antecedent.Add(inter);
antecedent.Add(closeArc);
antecedent.Add(farArc);
newGrounded.Add(new EdgeAggregator(antecedent, gaaeq, annotation));
return newGrounded;
}
//
// A \
// \ B
// \ /
// O \/ X
// /\
// / \
// C / D
//
// Two Secants
// Intersection(X, AD, BC) -> 2 * Angle(AXC) = MajorArc(AC) - MinorArc(AC)
//
public static List<EdgeAggregator> InstantiateTwoSecantsTheorem(Intersection inter, Circle circle)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
// Is this intersection explicitly outside the circle? That is, the point of intersection exterior?
if (!circle.PointIsExterior(inter.intersect)) return newGrounded;
//
// Get the chords
//
Segment chord1 = circle.ContainsChord(inter.lhs);
Segment chord2 = circle.ContainsChord(inter.rhs);
if (chord1 == null || chord2 == null) return newGrounded;
Point closeChord1Pt = null;
Point closeChord2Pt = null;
Point farChord1Pt = null;
Point farChord2Pt = null;
if (Segment.Between(chord1.Point1, chord1.Point2, inter.intersect))
{
closeChord1Pt = chord1.Point1;
farChord1Pt = chord1.Point2;
}
else
{
closeChord1Pt = chord1.Point2;
farChord1Pt = chord1.Point1;
}
if (Segment.Between(chord2.Point1, chord2.Point2, inter.intersect))
{
closeChord2Pt = chord2.Point1;
farChord2Pt = chord2.Point2;
}
else
{
closeChord2Pt = chord2.Point2;
farChord2Pt = chord2.Point1;
}
//
// Get the arcs
//
Arc closeArc = Arc.GetFigureMinorArc(circle, closeChord1Pt, closeChord2Pt);
Arc farArc = Arc.GetFigureMinorArc(circle, farChord1Pt, farChord2Pt);
Angle theAngle = Angle.AcquireFigureAngle(new Angle(closeChord1Pt, inter.intersect, closeChord2Pt));
//
// Construct the new relationship
//
NumericValue two = new NumericValue(2);
//GeometricAngleEquation gaeq = new GeometricAngleEquation(new Multiplication(two, theAngle), new Subtraction(farArc, closeArc));
GeometricAngleArcEquation gaaeq = new GeometricAngleArcEquation(new Multiplication(two, theAngle), new Subtraction(farArc, closeArc));
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(inter);
antecedent.Add(circle);
antecedent.Add(closeArc);
antecedent.Add(farArc);
newGrounded.Add(new EdgeAggregator(antecedent, gaaeq, annotation));
return newGrounded;
}
//private static readonly string NAME = "Simplification";
//
// Given an equation, simplify algebraically using the following notions:
// A + A = B -> 2A = B
// A + B = B + C -> A = C
// A + B = 2B + C -> A = B + C
//
public static Equation Simplify(Equation original)
{
// Do we have an equation?
if (original == null) throw new ArgumentException();
// Is the equation 0 = 0? This should be allowed at it indicates a tautology
if (original.lhs.Equals(new NumericValue(0)) && original.rhs.Equals(new NumericValue(0)))
{
throw new ArgumentException("Should not have an equation that is 0 = 0: " + original.ToString());
}
//
// Ideally, flattening would:
// Remove all subtractions -> adding a negative instead
// Distribute subtraction or multiplication over addition
//
// Flatten the equation so that each side is a sum of atomic expressions
Equation copyEq = (Equation)original.DeepCopy();
FlatEquation flattened = new FlatEquation(copyEq.lhs.CollectTerms(), copyEq.rhs.CollectTerms());
//Debug.WriteLine("Equation prior to simplification: " + flattened.ToString());
// Combine terms only on each side (do not cross =)
FlatEquation combined = CombineLikeTerms(flattened);
//Debug.WriteLine("Equation after like terms combined on both sides: " + combined);
// Combine terms across the equal sign
FlatEquation across = CombineLikeTermsAcrossEqual(combined);
//Debug.WriteLine("Equation after simplifying both sides: " + across);
FlatEquation constSimplify = SimplifyForMultipliersAndConstants(across);
//
// Inflate the equation
//
Equation inflated = null;
GroundedClause singleLeftExp = InflateEntireSide(constSimplify.lhsExps);
GroundedClause singleRightExp = InflateEntireSide(constSimplify.rhsExps);
if (original is AlgebraicSegmentEquation)
{
inflated = new AlgebraicSegmentEquation(singleLeftExp, singleRightExp);
}
else if (original is GeometricSegmentEquation)
{
inflated = new GeometricSegmentEquation(singleLeftExp, singleRightExp);
}
else if (original is AlgebraicAngleEquation)
{
inflated = new AlgebraicAngleEquation(singleLeftExp, singleRightExp);
}
else if (original is GeometricAngleEquation)
{
inflated = new GeometricAngleEquation(singleLeftExp, singleRightExp);
}
else if (original is AlgebraicArcEquation)
{
inflated = new AlgebraicArcEquation(singleLeftExp, singleRightExp);
}
else if (original is GeometricArcEquation)
{
inflated = new GeometricArcEquation(singleLeftExp, singleRightExp);
}
else if (original is AlgebraicAngleArcEquation)
{
inflated = new AlgebraicAngleArcEquation(singleLeftExp, singleRightExp);
}
else if (original is GeometricAngleArcEquation)
{
inflated = new GeometricAngleArcEquation(singleLeftExp, singleRightExp);
}
// If simplifying didn't do anything, return the original equation
if (inflated.Equals(original))
{
return original;
}
//
// 0 = 0 should not be allowable.
//
if (inflated.lhs.Equals(new NumericValue(0)) && inflated.rhs.Equals(new NumericValue(0)))
{
return null;
}
return inflated;
}
//
// A \ / B
// \ /
// \ /
// O \/ X
// /\
// / \
// C / \ D
//
// Intersection(Segment(AD), Segment(BC)) -> 2 * Angle(CXA) = Arc(A, C) + Arc(B, D),
// 2 * Angle(BXD) = Arc(A, C) + Arc(B, D),
// 2 * Angle(AXB) = Arc(A, B) + Arc(C, D),
// 2 * Angle(CXD) = Arc(A, B) + Arc(C, D)
//
public static List<EdgeAggregator> InstantiateTheorem(Intersection inter, Circle circle)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
// Is this intersection explicitly inside the circle? That is, the point of intersection interior?
if (!circle.PointIsInterior(inter.intersect)) return newGrounded;
//
// Get the chords
//
Segment chord1 = circle.GetChord(inter.lhs);
Segment chord2 = circle.GetChord(inter.rhs);
if (chord1 == null || chord2 == null) return newGrounded;
//
// Group 1
//
Arc arc1 = Arc.GetFigureMinorArc(circle, chord1.Point1, chord2.Point1);
Angle angle1 = Angle.AcquireFigureAngle(new Angle(chord1.Point1, inter.intersect, chord2.Point1));
Arc oppArc1 = Arc.GetFigureMinorArc(circle, chord1.Point2, chord2.Point2);
Angle oppAngle1 = Angle.AcquireFigureAngle(new Angle(chord1.Point2, inter.intersect, chord2.Point2));
//
// Group 2
//
Arc arc2 = Arc.GetFigureMinorArc(circle, chord1.Point1, chord2.Point2);
Angle angle2 = Angle.AcquireFigureAngle(new Angle(chord1.Point1, inter.intersect, chord2.Point2));
Arc oppArc2 = Arc.GetFigureMinorArc(circle, chord1.Point2, chord2.Point1);
Angle oppAngle2 = Angle.AcquireFigureAngle(new Angle(chord1.Point2, inter.intersect, chord2.Point1));
//
// Construct each of the 4 equations.
//
NumericValue two = new NumericValue(2);
GeometricAngleArcEquation gaeq1 = new GeometricAngleArcEquation(new Multiplication(two, angle1), new Addition(arc1, oppArc1));
GeometricAngleArcEquation gaeq2 = new GeometricAngleArcEquation(new Multiplication(two, oppAngle1), new Addition(arc1, oppArc1));
GeometricAngleArcEquation gaeq3 = new GeometricAngleArcEquation(new Multiplication(two, angle2), new Addition(arc2, oppArc2));
GeometricAngleArcEquation gaeq4 = new GeometricAngleArcEquation(new Multiplication(two, oppAngle2), new Addition(arc2, oppArc2));
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(inter);
antecedent.Add(circle);
newGrounded.Add(new EdgeAggregator(antecedent, gaeq1, annotation));
newGrounded.Add(new EdgeAggregator(antecedent, gaeq2, annotation));
newGrounded.Add(new EdgeAggregator(antecedent, gaeq3, annotation));
newGrounded.Add(new EdgeAggregator(antecedent, gaeq4, annotation));
return newGrounded;
}
private static EdgeAggregator CreateClause(Intersection inter, GroundedClause original, Angle theAngle, Arc theArc)
{
Multiplication product = new Multiplication(new NumericValue(2), theAngle);
GeometricAngleArcEquation angArcEq = new GeometricAngleArcEquation(product, theArc);
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(original);
antecedent.Add(inter);
return new EdgeAggregator(antecedent, angArcEq, annotation);
}
//
// C
// /)
// / )
// / )
// / )
// A /)_________ B
//
// Tangent(Circle(O), Segment(AB)), Intersection(Segment(AC), Segment(AB)) -> 2 * Angle(CAB) = Arc(C, B)
//
public static List<EdgeAggregator> InstantiateTheorem(Intersection inter, Tangent tangent, GroundedClause original)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
CircleSegmentIntersection tan = tangent.intersection as CircleSegmentIntersection;
//
// Does this tangent apply to this intersection?
//
if (!inter.intersect.StructurallyEquals(tangent.intersection.intersect)) return newGrounded;
Segment secant = null;
Segment tanSegment = null;
if (tan.HasSegment(inter.lhs))
{
secant = inter.rhs;
tanSegment = inter.lhs;
}
else if (tan.HasSegment(inter.rhs))
{
secant = inter.lhs;
tanSegment = inter.rhs;
}
else return newGrounded;
//
// Acquire the angle and intercepted arc.
//
Segment chord = tan.theCircle.GetChord(secant);
if (chord == null) return newGrounded;
//Segment chord = tan.theCircle.ContainsChord(secant);
// Arc
// We want the MINOR ARC only!
if (tan.theCircle.DefinesDiameter(chord))
{
Arc theArc = null;
Point midpt = PointFactory.GeneratePoint(tan.theCircle.Midpoint(chord.Point1, chord.Point2));
Point opp = PointFactory.GeneratePoint(tan.theCircle.OppositePoint(midpt));
Point tanPoint = tanSegment.OtherPoint(inter.intersect);
if (tanPoint != null)
{
// Angle; the smaller angle is always the chosen angle
Angle theAngle = new Angle(chord.OtherPoint(inter.intersect), inter.intersect, tanPoint);
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, midpt, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, opp, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
}
else
{
// Angle; the smaller angle is always the chosen angle
Angle theAngle = new Angle(chord.OtherPoint(inter.intersect), inter.intersect, tanSegment.Point1);
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, midpt, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, opp, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
// Angle; the smaller angle is always the chosen angle
theAngle = new Angle(chord.OtherPoint(inter.intersect), inter.intersect, tanSegment.Point2);
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, midpt, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
theArc = new Semicircle(tan.theCircle, chord.Point1, chord.Point2, opp, chord);
newGrounded.Add(CreateClause(inter, original, theAngle, theArc));
}
}
else
{
Arc theArc = new MinorArc(tan.theCircle, chord.Point1, chord.Point2);
// Angle; the smaller angle is always the chosen angle
Point endPnt = (inter.intersect.StructurallyEquals(tanSegment.Point1)) ? tanSegment.Point2 : tanSegment.Point1;
Angle theAngle = new Angle(chord.OtherPoint(inter.intersect), inter.intersect, endPnt);
if (theAngle.measure > 90)
{
//If the angle endpoint was already set to Point2, or if the intersect equals Point2, then the smaller angle does not exist
//In this case, should we create a major arc or return nothing?
if (endPnt.StructurallyEquals(tanSegment.Point2) || inter.intersect.StructurallyEquals(tanSegment.Point2)) return newGrounded;
theAngle = new Angle(chord.OtherPoint(inter.intersect), inter.intersect, tanSegment.Point2);
}
Multiplication product = new Multiplication(new NumericValue(2), theAngle);
GeometricAngleArcEquation angArcEq = new GeometricAngleArcEquation(product, theArc);
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(original);
antecedent.Add(inter);
antecedent.Add(theArc);
antecedent.Add(theAngle);
newGrounded.Add(new EdgeAggregator(antecedent, angArcEq, annotation));
}
return newGrounded;
}
// / A
// / |)
// / | )
// / | )
// Q------- O---X---) D
// \ | )
// \ | )
// \ |)
// \ C
//
// Inscribed Angle(AQC) -> Angle(AQC) = 2 * Arc(AC)
//
private static List<EdgeAggregator> InstantiateTheorem(Circle circle, Angle angle)
{
List<EdgeAggregator> newGrounded = new List<EdgeAggregator>();
// Acquire all circles in which the angle is inscribed
List<Circle> circles = Circle.IsInscribedAngle(angle);
//
// Get this particular inscribed circle.
//
Circle inscribed = null;
foreach (Circle c in circles)
{
if (circle.StructurallyEquals(c)) inscribed = c;
}
if (inscribed == null) return newGrounded;
// Get the intercepted arc
Arc intercepted = Arc.GetInterceptedArc(inscribed, angle);
//
// Create the equation
//
Multiplication product = new Multiplication(new NumericValue(2), angle);
GeometricAngleArcEquation gaaeq = new GeometricAngleArcEquation(product, intercepted);
// For hypergraph
List<GroundedClause> antecedent = new List<GroundedClause>();
antecedent.Add(circle);
antecedent.Add(angle);
antecedent.Add(intercepted);
newGrounded.Add(new EdgeAggregator(antecedent, gaaeq, annotation));
return newGrounded;
}