//Demonstrates: congruent chords have congruent arcs
public Page306Theorem7_4_1_Semicircle(bool onoff, bool complete) : base(onoff, complete)
{
Point o = new Point("O", 0, 0); points.Add(o);
Point c = new Point("C", 15, 0); points.Add(c);
Point s = new Point("S", -3, -4); points.Add(s);
Point t = new Point("T", 3, 4); points.Add(t);
Point u = new Point("U", 2, Math.Sqrt(21)); points.Add(u);
Point a = new Point("A", 12, 4); points.Add(a);
Point b = new Point("B", 18, -4); points.Add(b);
Point d = new Point("D", 16, -Math.Sqrt(24)); points.Add(d);
Segment st = new Segment(s, t); segments.Add(st);
Segment ab = new Segment(a, b); segments.Add(ab);
Segment ou = new Segment(o, u); segments.Add(ou);
Segment cd = new Segment(c, d); segments.Add(cd);
Circle c1 = new Circle(o, 5.0);
Circle c2 = new Circle(c, 5.0);
circles.Add(c1);
circles.Add(c2);
parser = new LiveGeometry.TutorParser.HardCodedParserMain(points, collinear, segments, circles, onoff);
given.Add(new GeometricCongruentSegments(st, ab));
Semicircle semi1 = (Semicircle)parser.Get(new Semicircle(c1, s, t, u, st));
Semicircle semi2 = (Semicircle)parser.Get(new Semicircle(c2, a, b, d, ab));
goals.Add(new GeometricCongruentArcs(semi1, semi2));
}
//
// C
// |\
// | \
// | \
// | O
// | \
// |_ \
// A |_|____\ B
//
// SemiCircle(O, BC), Angle(BAC) -> RightAngle(BAC)
//
public static List <EdgeAggregator> InstantiateTheorem(Semicircle semi, Angle angle, GroundedClause original)
{
List <EdgeAggregator> newGrounded = new List <EdgeAggregator>();
// The angle needs to be inscribed in the given semicircle
// Note: Previously this was checked indirectly by verifying that the angle intercepts a semicircle, but since semicircles now
// require 3 points to be defined, it is safer to directly verify that the angle is inscribed in the semicircle.
// (There may not have been any points defined on the other side of the diameter,
// meaning there would not actually be any defined semicircles which the angle intercepts).
if (!semi.AngleIsInscribed(angle))
{
return(newGrounded);
}
Strengthened newRight = new Strengthened(angle, new RightAngle(angle));
// For hypergraph
List <GroundedClause> antecedent = new List <GroundedClause>();
antecedent.Add(original);
antecedent.Add(angle);
newGrounded.Add(new EdgeAggregator(antecedent, newRight, annotation));
return(newGrounded);
}
// Construct the region between a circle and circle:
// __
// ( (
// ( (
// ( (
// ( (
// ( (
// --
private Atomizer.AtomicRegion ConstructBasicCircleCircleRegion(Segment chord, Circle smaller, Circle larger)
{
AtomicRegion region = new AtomicRegion();
Arc arc1 = null;
if (smaller.DefinesDiameter(chord))
{
Point midpt = smaller.Midpoint(chord.Point1, chord.Point2, larger.Midpoint(chord.Point1, chord.Point2));
arc1 = new Semicircle(smaller, chord.Point1, chord.Point2, midpt, chord);
}
else
{
arc1 = new MinorArc(smaller, chord.Point1, chord.Point2);
}
MinorArc arc2 = new MinorArc(larger, chord.Point1, chord.Point2);
region.AddConnection(chord.Point1, chord.Point2, ConnectionType.ARC, arc1);
region.AddConnection(chord.Point1, chord.Point2, ConnectionType.ARC, arc2);
return(region);
}
//
// If one of the endpoints of that is inside of this; and vice versa.
//
public bool Overlap(Connection that)
{
if (that.type != this.type)
{
return(false);
}
if (this.type == ConnectionType.ARC)
{
if (!(this.segmentOrArc as Arc).StructurallyEquals(this.segmentOrArc as Arc))
{
return(false);
}
// If the arcs just touch, it's not overlap.
if (this.segmentOrArc is MinorArc)
{
MinorArc minor = this.segmentOrArc as MinorArc;
if (minor.PointLiesStrictlyOn(that.endpoint1) || minor.PointLiesStrictlyOn(that.endpoint2))
{
return(true);
}
}
else if (this.segmentOrArc is Semicircle)
{
Semicircle semi = this.segmentOrArc as Semicircle;
if (semi.PointLiesStrictlyOn(that.endpoint1) || semi.PointLiesStrictlyOn(that.endpoint2))
{
return(true);
}
}
else if (this.segmentOrArc is MajorArc)
{
MajorArc major = this.segmentOrArc as MajorArc;
if (major.PointLiesStrictlyOn(that.endpoint1) || major.PointLiesStrictlyOn(that.endpoint2))
{
return(true);
}
}
return(false);
}
else if (this.type == ConnectionType.SEGMENT)
{
Segment thisSegment = this.segmentOrArc as Segment;
Segment thatSegment = that.segmentOrArc as Segment;
if (!thisSegment.IsCollinearWith(thatSegment))
{
return(false);
}
return(thisSegment.CoincidingWithOverlap(thatSegment));
}
return(false);
}
private List <Atomizer.AtomicRegion> ConvertToSemicircle(Segment diameter, Semicircle semi)
{
// Verification Step 2.
if (!diameter.PointLiesOnAndExactlyBetweenEndpoints(semi.theCircle.center))
{
throw new Exception("Semicircle: expected center between endpoints.");
}
Sector sector = new Sector(semi);
return(Utilities.MakeList <AtomicRegion>(new ShapeAtomicRegion(sector)));
}
private Semicircle CreateImpliedSemicircle(Circle circle, Segment diameter, Point oppositePnt)
{
Point midpt = circle.Midpoint(diameter.Point1, diameter.Point2);
//Create semicircles from the midpt and the given oppositePnt, make sure they do not form the same side
Semicircle semi1 = new Semicircle(circle, diameter.Point1, diameter.Point2, midpt, diameter);
if (semi1.SameSideSemicircle(new Semicircle(circle, diameter.Point1, diameter.Point2, oppositePnt, diameter)))
{
semi1 = new Semicircle(circle, diameter.Point1, diameter.Point2, circle.OppositePoint(midpt), diameter);
}
return(semi1);
}
//
// C
// |\
// | \
// | \
// | O
// | \
// |_ \
// A |_|____\ B
//
// SemiCircle(O, BC), Angle(BAC) -> RightAngle(BAC)
//
public static List <EdgeAggregator> Instantiate(GroundedClause clause)
{
annotation.active = EngineUIBridge.JustificationSwitch.ANGLE_INSCRIBED_SEMICIRCLE_IS_RIGHT;
List <EdgeAggregator> newGrounded = new List <EdgeAggregator>();
if (clause is Angle)
{
Angle angle = clause as Angle;
foreach (Semicircle semi in candidateSemiCircles)
{
newGrounded.AddRange(InstantiateTheorem(semi, angle, semi));
}
foreach (Strengthened streng in candidateStrengthened)
{
newGrounded.AddRange(InstantiateTheorem(streng.strengthened as Semicircle, angle, streng));
}
candidateInscribed.Add(angle);
}
else if (clause is Semicircle)
{
Semicircle semi = clause as Semicircle;
foreach (Angle angle in candidateInscribed)
{
newGrounded.AddRange(InstantiateTheorem(semi, angle, semi));
}
candidateSemiCircles.Add(semi);
}
else if (clause is Strengthened)
{
Strengthened streng = clause as Strengthened;
if (!(streng.strengthened is Semicircle))
{
return(newGrounded);
}
foreach (Angle angle in candidateInscribed)
{
newGrounded.AddRange(InstantiateTheorem(streng.strengthened as Semicircle, angle, streng));
}
candidateStrengthened.Add(streng);
}
return(newGrounded);
}
private void AddSemicircleClauses(Semicircle semi)
{
if (!GeometryTutorLib.Utilities.HasStructurally <Semicircle>(semiCircles, semi))
{
semiCircles.Add(semi);
semicircleSectors.Add(new Sector(semi));
}
//Add arcInMiddle
//For semicircles, only considering the defining middle point as an inMiddle point
//This is to avoid arc equations such as MinorArc(RX) + MinorArc(XT) = Semicircle(RST), which might not make sense to a user
GeometryTutorLib.Utilities.AddStructurallyUnique <ArcInMiddle>(arcInMiddle, new ArcInMiddle(semi.middlePoint, semi));
}
// Construct the region between a chord and the circle arc:
// (|
// ( |
// ( |
// ( |
// (|
//
private List <AtomicRegion> ConstructBasicLineCircleRegion(Segment chord, Circle circle)
{
//
// Standard
//
if (!circle.DefinesDiameter(chord))
{
AtomicRegion region = new AtomicRegion();
Arc theArc = new MinorArc(circle, chord.Point1, chord.Point2);
region.AddConnection(chord.Point1, chord.Point2, ConnectionType.ARC, theArc);
region.AddConnection(chord.Point1, chord.Point2, ConnectionType.SEGMENT, chord);
return(Utilities.MakeList <AtomicRegion>(region));
}
//
// Semi-circles
//
Point midpt = circle.Midpoint(chord.Point1, chord.Point2);
Arc semi1 = new Semicircle(circle, chord.Point1, chord.Point2, midpt, chord);
ShapeAtomicRegion region1 = new ShapeAtomicRegion(new Sector(semi1));
Point opp = circle.OppositePoint(midpt);
Arc semi2 = new Semicircle(circle, chord.Point1, chord.Point2, opp, chord);
ShapeAtomicRegion region2 = new ShapeAtomicRegion(new Sector(semi2));
List <AtomicRegion> regions = new List <AtomicRegion>();
regions.Add(region1);
regions.Add(region2);
return(regions);
}
private List <Atomizer.AtomicRegion> MixedArcChordedRegion(List <Circle> thatCircles, UndirectedPlanarGraph.PlanarGraph graph)
{
List <AtomicRegion> regions = new List <AtomicRegion>();
// Every segment may be have a set of circles. (on each side) surrounding it.
// Keep parallel lists of: (1) segments, (2) (real) arcs, (3) left outer circles, and (4) right outer circles
Segment[] regionsSegments = new Segment[points.Count];
Arc[] arcSegments = new Arc[points.Count];
Circle[] leftOuterCircles = new Circle[points.Count];
Circle[] rightOuterCircles = new Circle[points.Count];
//
// Populate the parallel arrays.
//
int currCounter = 0;
for (int p = 0; p < points.Count;)
{
UndirectedPlanarGraph.PlanarGraphEdge edge = graph.GetEdge(points[p], points[(p + 1) % points.Count]);
Segment currSegment = new Segment(points[p], points[(p + 1) % points.Count]);
//
// If a known segment, seek a sequence of collinear segments.
//
if (edge.edgeType == UndirectedPlanarGraph.EdgeType.REAL_SEGMENT)
{
Segment actualSeg = currSegment;
bool collinearExists = false;
int prevPtIndex;
for (prevPtIndex = p + 1; prevPtIndex < points.Count; prevPtIndex++)
{
// Make another segment with the next point.
Segment nextSeg = new Segment(points[p], points[(prevPtIndex + 1) % points.Count]);
// CTA: This criteria seems invalid in some cases....; may not have collinearity
// We hit the end of the line of collinear segments.
if (!currSegment.IsCollinearWith(nextSeg))
{
break;
}
collinearExists = true;
actualSeg = nextSeg;
}
// If there exists an arc over the actual segment, we have an embedded circle to consider.
regionsSegments[currCounter] = actualSeg;
if (collinearExists)
{
UndirectedPlanarGraph.PlanarGraphEdge collEdge = graph.GetEdge(actualSeg.Point1, actualSeg.Point2);
if (collEdge != null)
{
if (collEdge.edgeType == UndirectedPlanarGraph.EdgeType.REAL_ARC)
{
// Find all applicable circles
List <Circle> circles = GetAllApplicableCircles(thatCircles, actualSeg.Point1, actualSeg.Point2);
// Get the exact outer circles for this segment (and create any embedded regions).
regions.AddRange(ConvertToCircleCircle(actualSeg, circles, out leftOuterCircles[currCounter], out rightOuterCircles[currCounter]));
}
}
}
currCounter++;
p = prevPtIndex;
}
else if (edge.edgeType == UndirectedPlanarGraph.EdgeType.REAL_DUAL)
{
regionsSegments[currCounter] = new Segment(points[p], points[(p + 1) % points.Count]);
// Get the exact chord and set of circles
Segment chord = regionsSegments[currCounter];
// Find all applicable circles
List <Circle> circles = GetAllApplicableCircles(thatCircles, points[p], points[(p + 1) % points.Count]);
// Get the exact outer circles for this segment (and create any embedded regions).
regions.AddRange(ConvertToCircleCircle(chord, circles, out leftOuterCircles[currCounter], out rightOuterCircles[currCounter]));
currCounter++;
p++;
}
else if (edge.edgeType == UndirectedPlanarGraph.EdgeType.REAL_ARC)
{
//
// Find the unique circle that contains these two points.
// (if more than one circle has these points, we would have had more intersections and it would be a direct chorded region)
//
List <Circle> circles = GetAllApplicableCircles(thatCircles, points[p], points[(p + 1) % points.Count]);
if (circles.Count != 1)
{
throw new Exception("Need ONLY 1 circle for REAL_ARC atom id; found (" + circles.Count + ")");
}
arcSegments[currCounter++] = new MinorArc(circles[0], points[p], points[(p + 1) % points.Count]);
p++;
}
}
//
// Check to see if this is a region in which some connections are segments and some are arcs.
// This means there were no REAL_DUAL edges.
//
List <AtomicRegion> generalRegions = GeneralAtomicRegion(regionsSegments, arcSegments);
if (generalRegions.Any())
{
return(generalRegions);
}
// Copy the segments into a list (ensuring no nulls)
List <Segment> actSegments = new List <Segment>();
foreach (Segment side in regionsSegments)
{
if (side != null)
{
actSegments.Add(side);
}
}
// Construct a polygon out of the straight-up segments
// This might be a polygon that defines a pathological region.
Polygon poly = Polygon.MakePolygon(actSegments);
// Determine which outermost circles apply inside of this polygon.
Circle[] circlesCutInsidePoly = new Circle[actSegments.Count];
for (int p = 0; p < actSegments.Count; p++)
{
if (leftOuterCircles[p] != null && rightOuterCircles[p] == null)
{
circlesCutInsidePoly[p] = CheckCircleCutInsidePolygon(poly, leftOuterCircles[p], actSegments[p].Point1, actSegments[p].Point2);
}
else if (leftOuterCircles[p] == null && rightOuterCircles[p] != null)
{
circlesCutInsidePoly[p] = CheckCircleCutInsidePolygon(poly, rightOuterCircles[p], actSegments[p].Point1, actSegments[p].Point2);
}
else if (leftOuterCircles[p] != null && rightOuterCircles[p] != null)
{
circlesCutInsidePoly[p] = CheckCircleCutInsidePolygon(poly, leftOuterCircles[p], actSegments[p].Point1, actSegments[p].Point2);
if (circlesCutInsidePoly[p] == null)
{
circlesCutInsidePoly[p] = rightOuterCircles[p];
}
}
else
{
circlesCutInsidePoly[p] = null;
}
}
bool isStrictPoly = true;
for (int p = 0; p < actSegments.Count; p++)
{
if (circlesCutInsidePoly[p] != null || arcSegments[p] != null)
{
isStrictPoly = false;
break;
}
}
// This is just a normal shape region: polygon.
if (isStrictPoly)
{
regions.Add(new ShapeAtomicRegion(poly));
}
// A circle cuts into the polygon.
else
{
//
// Now that all interior arcs have been identified, construct the atomic (probably pathological) region
//
AtomicRegion pathological = new AtomicRegion();
for (int p = 0; p < actSegments.Count; p++)
{
//
// A circle cutting inside the polygon
//
if (circlesCutInsidePoly[p] != null)
{
Arc theArc = null;
if (circlesCutInsidePoly[p].DefinesDiameter(regionsSegments[p]))
{
Point midpt = circlesCutInsidePoly[p].Midpoint(regionsSegments[p].Point1, regionsSegments[p].Point2);
if (!poly.IsInPolygon(midpt))
{
midpt = circlesCutInsidePoly[p].OppositePoint(midpt);
}
theArc = new Semicircle(circlesCutInsidePoly[p], regionsSegments[p].Point1, regionsSegments[p].Point2, midpt, regionsSegments[p]);
}
else
{
theArc = new MinorArc(circlesCutInsidePoly[p], regionsSegments[p].Point1, regionsSegments[p].Point2);
}
pathological.AddConnection(regionsSegments[p].Point1, regionsSegments[p].Point2, ConnectionType.ARC, theArc);
}
//
else
{
// We have a direct arc
if (arcSegments[p] != null)
{
pathological.AddConnection(regionsSegments[p].Point1, regionsSegments[p].Point2,
ConnectionType.ARC, arcSegments[p]);
}
// Use the segment
else
{
pathological.AddConnection(regionsSegments[p].Point1, regionsSegments[p].Point2,
ConnectionType.SEGMENT, regionsSegments[p]);
}
}
}
regions.Add(pathological);
}
return(regions);
}
//
// Detect diameters and generate all of the Semicircle Arc and ArcInMiddle clauses
//
private void GenerateSemicircleClauses(Circle circle)
{
if (circle.pointsOnCircle.Count == 2)
{
Segment diameter = new Segment(circle.pointsOnCircle[0], circle.pointsOnCircle[1]);
if (circle.DefinesDiameter(diameter))
{
Point midpt = circle.Midpoint(diameter.Point1, diameter.Point2);
Point opp = circle.OppositePoint(midpt);
AddSemicircleClauses(new Semicircle(circle, diameter.Point1, diameter.Point2, midpt, diameter));
AddSemicircleClauses(new Semicircle(circle, diameter.Point1, diameter.Point2, opp, diameter));
}
}
for (int p1 = 0; p1 < circle.pointsOnCircle.Count - 1; p1++)
{
for (int p2 = p1 + 1; p2 < circle.pointsOnCircle.Count; p2++)
{
Segment diameter = new Segment(circle.pointsOnCircle[p1], circle.pointsOnCircle[p2]);
if (circle.DefinesDiameter(diameter))
{
//Get the endpoints of the diameter and the indices of these endpoints
Point e1 = diameter.Point1;
Point e2 = diameter.Point2;
//int p1 = circle.pointsOnCircle.IndexOf(e1);
//int p2 = circle.pointsOnCircle.IndexOf(e2);
////For partitioning purposes, order of the endpoints matters. Make sure p1 holds the lower of the two indices
//if (p1 > p2)
//{
// int p3 = p1;
// p1 = p2;
// p2 = p3;
//}
// Partition the remaining points on the circle
List <Point> minorArcPoints;
List <Point> majorArcPoints;
PartitionSemiCircleArcPoints(circle.pointsOnCircle, p1, p2, out minorArcPoints, out majorArcPoints);
// Semicircle requires 3 points to be defined - the two endpoints and a point inbetween
// The minorArcPoints and majorArcPoints lists contain all the potential inbetween points for either side of the diameter
// Handle 'side' 1:
// If majorArcPoints is empty, create an implied semicircle (minorArcPoints should be guaranteed to have at least one point, since
// the case of having only 2 points on the circle was already handled)
if (majorArcPoints.Count == 0 && minorArcPoints.Count != 0)
{
AddSemicircleClauses(CreateImpliedSemicircle(circle, diameter, minorArcPoints[0]));
}
else
{
for (int i = 0; i < majorArcPoints.Count; ++i)
{
Semicircle semi = new Semicircle(circle, e1, e2, majorArcPoints[i], minorArcPoints, majorArcPoints, diameter);
AddSemicircleClauses(semi);
}
}
// Handle 'side' 2:
if (minorArcPoints.Count == 0 && majorArcPoints.Count != 0)
{
AddSemicircleClauses(CreateImpliedSemicircle(circle, diameter, majorArcPoints[0]));
}
else
{
for (int i = 0; i < minorArcPoints.Count; ++i)
{
Semicircle semi = new Semicircle(circle, e1, e2, minorArcPoints[i], majorArcPoints, minorArcPoints, diameter);
AddSemicircleClauses(semi);
}
}
}
}
}
}
//
// 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);
}
