Theorem: Take any triangle, and draw , 2, -sectors at each vertex. The Generalized Morley triangle has as its vertices the points of intersecti\ on of 2, -sectors closest to the edges emanating from the two vertices of each edge These are concurrent. This ends this cute article that took, 0.017, seconds to generate. --------------------------------------------- Theorem: Take any triangle, and draw , 3, -sectors at each vertex. The Generalized Morley triangle has as its vertices the points of intersection of 3-sectors closest to the edges emanating from the two vertices of each edge Let , A12, A23, A31, be the DISTANCES-SQUARED, of that new triangle divided by the square of the length of the base( i.e. the side closest to th\ e intersection of edges A12 and A31 of the generalized Morley triangle. We have the following polynomial relations, of degree, 1, relating them. A12-A31 = 0 A23-A31 = 0 This ends this cute article that took, 0.152, seconds to generate. --------------------------------------------- Theorem: Take any triangle, and draw , 4, -sectors at each vertex. The Generalized Morley triangle has as its vertices the points of intersection of 4-sectors closest to the edges emanating from the two vertices of each edge Let , A12, A23, A31, be the DISTANCES-SQUARED, of that new triangle divided by the square of the length of the base( i.e. the side closest to the intersection of edges A12 and A31 of the generalized Morley triangle. We have the following polynomial relation, of degree, 14, relating them. -8*A12^14+88*A12^13*A23+32*A12^13*A31-516*A12^12*A23^2-200*A12^12*A23*A31-88* A12^12*A31^2+2016*A12^11*A23^3+512*A12^11*A23^2*A31+464*A12^11*A23*A31^2+192* A12^11*A31^3-5714*A12^10*A23^4+112*A12^10*A23^3*A31-1704*A12^10*A23^2*A31^2-688 *A12^10*A23*A31^3-328*A12^10*A31^4+12230*A12^9*A23^5-5184*A12^9*A23^4*A31+5552* A12^9*A23^3*A31^2+1344*A12^9*A23^2*A31^3+776*A12^9*A23*A31^4+480*A12^9*A31^5-\ 20113*A12^8*A23^6+19294*A12^8*A23^5*A31-17218*A12^8*A23^4*A31^2+1440*A12^8*A23^ 3*A31^3-2172*A12^8*A23^2*A31^4-664*A12^8*A23*A31^5-600*A12^8*A31^6+25584*A12^7* A23^7-41732*A12^7*A23^6*A31+43656*A12^7*A23^5*A31^2-16736*A12^7*A23^4*A31^3+ 6272*A12^7*A23^3*A31^4+1472*A12^7*A23^2*A31^5+224*A12^7*A23*A31^6+640*A12^7*A31 ^7-25192*A12^6*A23^8+61552*A12^6*A23^7*A31-82472*A12^6*A23^6*A31^2+55736*A12^6* A23^5*A31^3-25132*A12^6*A23^4*A31^4+4064*A12^6*A23^3*A31^5-1968*A12^6*A23^2*A31 ^6+224*A12^6*A23*A31^7-600*A12^6*A31^8+19108*A12^5*A23^9-64940*A12^5*A23^8*A31+ 112656*A12^5*A23^7*A31^2-110348*A12^5*A23^6*A31^3+68380*A12^5*A23^5*A31^4-23104 *A12^5*A23^4*A31^5+4064*A12^5*A23^3*A31^6+1472*A12^5*A23^2*A31^7-664*A12^5*A23* A31^8+480*A12^5*A31^9-11018*A12^4*A23^10+49596*A12^4*A23^9*A31-110136*A12^4*A23 ^8*A31^2+145168*A12^4*A23^7*A31^3-123950*A12^4*A23^6*A31^4+68380*A12^4*A23^5* A31^5-25132*A12^4*A23^4*A31^6+6272*A12^4*A23^3*A31^7-2172*A12^4*A23^2*A31^8+776 *A12^4*A23*A31^9-328*A12^4*A31^10+4704*A12^3*A23^11-27100*A12^3*A23^10*A31+ 75784*A12^3*A23^9*A31^2-128920*A12^3*A23^8*A31^3+145168*A12^3*A23^7*A31^4-\ 110348*A12^3*A23^6*A31^5+55736*A12^3*A23^5*A31^6-16736*A12^3*A23^4*A31^7+1440* A12^3*A23^3*A31^8+1344*A12^3*A23^2*A31^9-688*A12^3*A23*A31^10+192*A12^3*A31^11-\ 1414*A12^2*A23^12+10176*A12^2*A23^11*A31-35320*A12^2*A23^10*A31^2+75784*A12^2* A23^9*A31^3-110136*A12^2*A23^8*A31^4+112656*A12^2*A23^7*A31^5-82472*A12^2*A23^6 *A31^6+43656*A12^2*A23^5*A31^7-17218*A12^2*A23^4*A31^8+5552*A12^2*A23^3*A31^9-\ 1704*A12^2*A23^2*A31^10+464*A12^2*A23*A31^11-88*A12^2*A31^12+270*A12*A23^13-\ 2388*A12*A23^12*A31+10176*A12*A23^11*A31^2-27100*A12*A23^10*A31^3+49596*A12*A23 ^9*A31^4-64940*A12*A23^8*A31^5+61552*A12*A23^7*A31^6-41732*A12*A23^6*A31^7+ 19294*A12*A23^5*A31^8-5184*A12*A23^4*A31^9+112*A12*A23^3*A31^10+512*A12*A23^2* A31^11-200*A12*A23*A31^12+32*A12*A31^13-25*A23^14+270*A23^13*A31-1414*A23^12* A31^2+4704*A23^11*A31^3-11018*A23^10*A31^4+19108*A23^9*A31^5-25192*A23^8*A31^6+ 25584*A23^7*A31^7-20113*A23^6*A31^8+12230*A23^5*A31^9-5714*A23^4*A31^10+2016* A23^3*A31^11-516*A23^2*A31^12+88*A23*A31^13-8*A31^14+288*A12^13-2316*A12^12*A23 -1596*A12^12*A31+10344*A12^11*A23^2+8688*A12^11*A23*A31+4572*A12^11*A31^2-32278 *A12^10*A23^3-25926*A12^10*A23^2*A31-16152*A12^10*A23*A31^2-9220*A12^10*A31^3+ 76434*A12^9*A23^4+46466*A12^9*A23^3*A31+36766*A12^9*A23^2*A31^2+21104*A12^9*A23 *A31^3+12708*A12^9*A31^4-141044*A12^8*A23^5-39170*A12^8*A23^4*A31-61454*A12^8* A23^3*A31^2-45028*A12^8*A23^2*A31^3-10932*A12^8*A23*A31^4-11656*A12^8*A31^5+ 202428*A12^7*A23^6-38328*A12^7*A23^5*A31+108832*A12^7*A23^4*A31^2+59096*A12^7* A23^3*A31^3+17244*A12^7*A23^2*A31^4-2144*A12^7*A23*A31^5+4904*A12^7*A31^6-\ 224024*A12^6*A23^7+183892*A12^6*A23^6*A31-223264*A12^6*A23^5*A31^2-20584*A12^6* A23^4*A31^3-5852*A12^6*A23^3*A31^4-16952*A12^6*A23^2*A31^5+11696*A12^6*A23*A31^ 6+4904*A12^6*A31^7+189360*A12^5*A23^8-309268*A12^5*A23^7*A31+394056*A12^5*A23^6 *A31^2-135624*A12^5*A23^5*A31^3+31864*A12^5*A23^4*A31^4+34636*A12^5*A23^3*A31^5 -16952*A12^5*A23^2*A31^6-2144*A12^5*A23*A31^7-11656*A12^5*A31^8-120560*A12^4* A23^9+318912*A12^4*A23^8*A31-514504*A12^4*A23^7*A31^2+401896*A12^4*A23^6*A31^3-\ 206552*A12^4*A23^5*A31^4+31864*A12^4*A23^4*A31^5-5852*A12^4*A23^3*A31^6+17244* A12^4*A23^2*A31^7-10932*A12^4*A23*A31^8+12708*A12^4*A31^9+56412*A12^3*A23^10-\ 218600*A12^3*A23^9*A31+459916*A12^3*A23^8*A31^2-546072*A12^3*A23^7*A31^3+401896 *A12^3*A23^6*A31^4-135624*A12^3*A23^5*A31^5-20584*A12^3*A23^4*A31^6+59096*A12^3 *A23^3*A31^7-45028*A12^3*A23^2*A31^8+21104*A12^3*A23*A31^9-9220*A12^3*A31^10-\ 18514*A12^2*A23^11+99410*A12^2*A23^10*A31-274400*A12^2*A23^9*A31^2+459916*A12^2 *A23^8*A31^3-514504*A12^2*A23^7*A31^4+394056*A12^2*A23^6*A31^5-223264*A12^2*A23 ^5*A31^6+108832*A12^2*A23^4*A31^7-61454*A12^2*A23^3*A31^8+36766*A12^2*A23^2*A31 ^9-16152*A12^2*A23*A31^10+4572*A12^2*A31^11+3870*A12*A23^12-27838*A12*A23^11* A31+99410*A12*A23^10*A31^2-218600*A12*A23^9*A31^3+318912*A12*A23^8*A31^4-309268 *A12*A23^7*A31^5+183892*A12*A23^6*A31^6-38328*A12*A23^5*A31^7-39170*A12*A23^4* A31^8+46466*A12*A23^3*A31^9-25926*A12*A23^2*A31^10+8688*A12*A23*A31^11-1596*A12 *A31^12-400*A23^13+3870*A23^12*A31-18514*A23^11*A31^2+56412*A23^10*A31^3-120560 *A23^9*A31^4+189360*A23^8*A31^5-224024*A23^7*A31^6+202428*A23^6*A31^7-141044* A23^5*A31^8+76434*A23^4*A31^9-32278*A23^3*A31^10+10344*A23^2*A31^11-2316*A23* A31^12+288*A31^13-4050*A12^12+26460*A12^11*A23+27270*A12^11*A31-90918*A12^10* A23^2-121842*A12^10*A23*A31-103257*A12^10*A31^2+208020*A12^9*A23^3+291276*A12^9 *A23^2*A31+326826*A12^9*A23*A31^2+262038*A12^9*A31^3-353360*A12^8*A23^4-427172* A12^8*A23^3*A31-641694*A12^8*A23^2*A31^2-526040*A12^8*A23*A31^3-493082*A12^8* A31^4+468104*A12^7*A23^5+409052*A12^7*A23^4*A31+796476*A12^7*A23^3*A31^2+833240 *A12^7*A23^2*A31^3+542876*A12^7*A23*A31^4+709668*A12^7*A31^5-496388*A12^6*A23^6 -190420*A12^6*A23^5*A31-845150*A12^6*A23^4*A31^2-571124*A12^6*A23^3*A31^3-\ 945628*A12^6*A23^2*A31^4-225752*A12^6*A23*A31^5-801270*A12^6*A31^6+423224*A12^5 *A23^7-94736*A12^5*A23^6*A31+747072*A12^5*A23^5*A31^2+333708*A12^5*A23^4*A31^3+ 366664*A12^5*A23^3*A31^4+874488*A12^5*A23^2*A31^5-225752*A12^5*A23*A31^6+709668 *A12^5*A31^7-288346*A12^4*A23^8+286576*A12^4*A23^7*A31-704988*A12^4*A23^6*A31^2 +269308*A12^4*A23^5*A31^3-726660*A12^4*A23^4*A31^4+366664*A12^4*A23^3*A31^5-\ 945628*A12^4*A23^2*A31^6+542876*A12^4*A23*A31^7-493082*A12^4*A31^8+153180*A12^3 *A23^9-291490*A12^3*A23^8*A31+565940*A12^3*A23^7*A31^2-406424*A12^3*A23^6*A31^3 +269308*A12^3*A23^5*A31^4+333708*A12^3*A23^4*A31^5-571124*A12^3*A23^3*A31^6+ 833240*A12^3*A23^2*A31^7-526040*A12^3*A23*A31^8+262038*A12^3*A31^9-60950*A12^2* A23^10+186630*A12^2*A23^9*A31-412377*A12^2*A23^8*A31^2+565940*A12^2*A23^7*A31^3 -704988*A12^2*A23^6*A31^4+747072*A12^2*A23^5*A31^5-845150*A12^2*A23^4*A31^6+ 796476*A12^2*A23^3*A31^7-641694*A12^2*A23^2*A31^8+326826*A12^2*A23*A31^9-103257 *A12^2*A31^10+16500*A12*A23^11-72700*A12*A23^10*A31+186630*A12*A23^9*A31^2-\ 291490*A12*A23^8*A31^3+286576*A12*A23^7*A31^4-94736*A12*A23^6*A31^5-190420*A12* A23^5*A31^6+409052*A12*A23^4*A31^7-427172*A12*A23^3*A31^8+291276*A12*A23^2*A31^ 9-121842*A12*A23*A31^10+27270*A12*A31^11-2500*A23^12+16500*A23^11*A31-60950*A23 ^10*A31^2+153180*A23^9*A31^3-288346*A23^8*A31^4+423224*A23^7*A31^5-496388*A23^6 *A31^6+468104*A23^5*A31^7-353360*A23^4*A31^8+208020*A23^3*A31^9-90918*A23^2*A31 ^10+26460*A23*A31^11-4050*A31^12 = 0 This ends this cute article that took, 71.439, seconds to generate. --------------------------------------------- This took , 71.625, seconds.