Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Let L be the Euler line of ABC and La, Lb, Lc the parallels to L through A', B', C', resp.
Which is the locus of P such that the reflections of La, Lb, Lc in BC, CA, AB (or equivalently in PA', PB', PC', resp) are concurrent?
The Euler line?
And which is the locus of the point of concurrence as P moves on the locus?
[César Lozada]:
> The Euler line?
Confirmed. (+ Linf)
If OP=t*OH, then the point of concurrence Q(t) is
Q(t) = ((9*R^2-2*SW)*t-3*R^2+SB+SC)*( S^2-3*SB*SC) : : (barys)
Q is the midpoint of P and X(110) and lies on the line { 30, 113, 1495, 1511, 1514, 1524, 1525, 1531, 1533, 1539, 1544, 1545, 1546, 1553, 1554, 1555, 1558, 1561, 1568, 2682, 2685, 2686, 3233, 3258, 5642, 6739, 10272, 10564, 11064, 11693, 11694, 13202, 13392, 13857, 14499, 14500, 14857, 15354}
ETC pairs (P,Q):
(2,5642), (3,1511), (4,113), (5,10272), (23,1495), (140,13392), (382,1539), (549,11694), (858,11064), (3146,13202), (3153,1568), (3524,11693), (7464,10564), (7471,3233), (10296,1531), (10989,13857), (11050,15354), (14807,14499), (14808,14500)
Examples:
Q ( X(20) ) = midpoint of X(20) and X(110)
= (-a^2+b^2+c^2)*(2*a^4-(b^2+c^ 2)*a^2-(b^2-c^2)^2)^2 : : (barys)
= 3*X(2)-5*X(15051), 3*X(3)-X(265), 3*X(3)-2*X(6699), 5*X(3)-X(12902), 11*X(3)-5*X(15027), 5*X(3)-3*X(15061), 3*X(125)-2*X(265), 3*X(125)-4*X(6699), X(125)+2*X(12121), 5*X(125)-2*X(12902), 11*X(125)-10*X(15027), 5*X(125)-6*X(15061), X(265)+3*X(12121), 5*X(265)-3*X(12902), 11*X(265)-15*X(15027), 5*X(265)-9*X(15061), 2*X(6699)+3*X(12121), 10*X(6699)-3*X(12902), 22*X(6699)-15*X(15027), 10*X(6699)-9*X(15061), 2*X(7687)-5*X(15051), X(10733)-5*X(15051), 5*X(12121)+X(12902), 11*X(12121)+5*X(15027), 5*X(12121)+3*X(15061), X(12902)-3*X(15061)
= on lines: {2, 7687}, {3, 125}, {4, 5972}, {5, 12295}, {20, 110}, {22, 13289}, {30, 113}, {36, 12896}, {51, 9826}, {52, 14708}, {69, 74}, {114, 7422}, {133, 4240}, {140, 10113}, {154, 11744}, {159, 2935}, {165, 13211}, {184, 4846}, {343, 8703}, {378, 3818}, {381, 12900}, {382, 14643}, {394, 399}, {511, 1986}, {516, 11720}, {548, 12041}, {550, 5562}, {631, 6723}, {632, 15088}, {648, 5667}, {974, 6467}, {1060, 12888}, {1099, 1354}, {1154, 14049}, {1204, 12118}, {1216, 7723}, {1300, 8754}, {1503, 5181}, {1657, 7728}, {1974, 15462}, {1993, 12227}, {2071, 12827}, {2407, 16075}, {2420, 6793}, {2693, 13494}, {2771, 3650}, {2781, 3313}, {2854, 15151}, {2968, 3916}, {2979, 12219}, {3028, 15326}, {3043, 13619}, {3070, 8998}, {3071, 13990}, {3146, 15020}, {3163, 9408}, {3184, 9033}, {3289, 6781}, {3448, 3522}, {3523, 15059}, {3524, 15081}, {3529, 10721}, {3530, 11801}, {3576, 11735}, {3629, 14831}, {3917, 12358}, {3937, 13369}, {4235, 15595}, {4299, 10088}, {4302, 10091}, {5054, 15042}, {5076, 15046}, {5085, 15118}, {5204, 12904}, {5217, 12903}, {5609, 12103}, {5655, 15681}, {5731, 7984}, {5907, 12292}, {6000, 12825}, {6101, 15332}, {6361, 7978}, {6560, 10819}, {6561, 10820}, {7722, 11412}, {8907, 11413}, {9140, 10304}, {9517, 14689}, {9529, 14697}, {9729, 11800}, {9730, 12236}, {10117, 11414}, {10257, 13851}, {10303, 15023}, {10620, 15696}, {10625, 11562}, {10628, 15644}, {10706, 11001}, {10984, 13198}, {11204, 11442}, {11430, 14389}, {11561, 13391}, {11723, 12699}, {11807, 12824}, {12028, 14595}, {12054, 12201}, {12086, 13419}, {12133, 15030}, {12228, 13352}, {12244, 14094}, {12261, 13624}, {12273, 15072}, {12661, 15941}, {12702, 12898}, {13346, 15463}, {13416, 15738}, {14683, 15054}, {14847, 15774}, {15160, 15461}, {15161, 15460}
= midpoint of X(i) and X(j) for these {i,j}: {3, 12121}, {20, 110}, {1657, 7728}, {3529, 10721}, {5655, 15681}, {6361, 7978}, {7722, 11412}, {10625, 11562}, {10706, 11001}, {12244, 14094}, {12702, 12898}, {14683, 15054}
= reflection of X(i) in X(j) for these (i,j): (4, 5972), (52, 14708), (12261, 13624)
= anticomplement of X(7687)
= complement of X(10733)
= X(125) of ABC-X3 reflections triangle
= X(5972) of anti-Euler triangle
= X(12295) of Johnson triangle
= X(15030) of anti-orthocentroidal triangle
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 10733, 7687), (3, 265, 6699), (3, 12902, 15061), (4, 15035, 5972), (113, 1511, 5642), (146, 6053, 15063), (265, 6699, 125), (376, 12383, 74), (616, 617, 1494), (1511, 1539, 10272), (1511, 13392, 11693), (1531, 11064, 1568), (1539, 10272, 113), (5642, 13202, 113), (10733, 15051, 2), (14499, 14500, 1568)
= [ 10.0744540867104600, 8.7101634648033260, -7.0391967260535190 ]
Q( X(21) ) = midpoint of X(21) and X(110)
= a*(a^3-(b+c)*a^2-(b^2+b*c+c^2) *a+(b+c)*(b^2+c^2))*(2*a^4-(b^ 2+c^2)*a^2-(b^2-c^2)^2)*(a+c)* (a+b) : : (barys)
= X(2948)+3*X(5426), X(3448)-5*X(15674), X(3651)-3*X(15035), X(5609)+2*X(12104), X(9140)-3*X(15671), X(9143)+3*X(15672), X(14683)+7*X(15676), 5*X(15040)-X(16117)
= on lines: {21, 104}, {30, 113}, {125, 6675}, {442, 5972}, {517, 12826}, {542, 15670}, {758, 11720}, {2948, 5426}, {3028, 5427}, {3448, 15674}, {3651, 15035}, {5428, 5663}, {5609, 12104}, {9140, 15671}, {9143, 15672}, {14683, 15676}, {15040, 16117}
= midpoint of X(21) and X(110)
= reflection of X(i) in X(j) for these (i,j): (125, 6675), (442, 5972)
= [ 2.0423649172083200, 0.6992631493379355, 2.2139292628850440 ]
Q(X(22)) = midpoint of X(22) and X(110)
= a^2*(-a^2+b^2+c^2)*(a^4-b^4+b^ 2*c^2-c^4)*(2*a^4-(b^2+c^2)*a^ 2-(b^2-c^2)^2) : : (barys)
= X(378)-3*X(15035), X(5609)+2*X(7555), X(12082)+5*X(15034)
= on lines: {3, 15738}, {22, 110}, {23, 6593}, {25, 15462}, {30, 113}, {69, 5648}, {125, 6676}, {182, 12099}, {184, 14984}, {343, 542}, {378, 15035}, {427, 5972}, {974, 2931}, {1370, 15131}, {1503, 12827}, {3549, 15133}, {3796, 5622}, {5422, 11746}, {5562, 5609}, {5663, 7502}, {7387, 16105}, {9140, 15080}, {9934, 12168}, {11598, 13445}, {12041, 14855}, {12082, 15034}, {12310, 13198}, {13394, 15760}
= midpoint of X(22) and X(110)
= reflection of X(i) in X(j) for these (i,j): (125, 6676), (427, 5972)
= [ -11.7279433314338800, -13.0347186784486000, 18.0775974122644300 ]
César Lozada
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