For any point {a^2 (u - v) (-u + w), b^2 (u - v) (v - w), c^2 (v - w) (-u + w)} on circumcircle of ABC, ellipse has equation:-3 a^4 c^4 (u-w)^2 (4 u^6+12 v^6-36 v^5 w+49 v^4 w^2-38 v^3 w^3+27 v^2 w^4-14 v w^5+4 w^6-2 u^5 (7 v+5 w)+u^4 (27 v^2+16 v w+17 w^2)-2 u^3 (19 v^3-3 v^2 w+19 v w^2+5 w^3)+u^2 (49 v^4-82 v^3 w+114 v^2 w^2-38 v w^3+17 w^4)-2 u (18 v^5-41 v^4 w+41 v^3 w^2-3 v^2 w^3-8 v w^4+5 w^5)) y^2+2 a^2 b^2 c^2 y (c^2 (20 u^8+20 v^8-74 v^7 w+191 v^6 w^2-290 v^5 w^3+305 v^4 w^4-248 v^3 w^5+176 v^2 w^6-80 v w^7+20 w^8-2 u^7 (43 v+37 w)+u^6 (233 v^2+136 v w+191 w^2)-2 u^5 (211 v^3+66 v^2 w+138 v w^2+145 w^3)+5 u^4 (106 v^4-2 v^3 w+69 v^2 w^2+46 v w^3+61 w^4)-2 u^3 (211 v^5+5 v^4 w-20 v^3 w^2+250 v^2 w^3-10 v w^4+124 w^5)+u^2 (233 v^6-132 v^5 w+345 v^4 w^2-500 v^3 w^3+750 v^2 w^4-312 v w^5+176 w^6)+u (-86 v^7+136 v^6 w-276 v^5 w^2+230 v^4 w^3+20 v^3 w^4-312 v^2 w^5+208 v w^6-80 w^7)) x+a^2 (20 u^8+20 v^8-86 v^7 w+233 v^6 w^2-422 v^5 w^3+530 v^4 w^4-422 v^3 w^5+233 v^2 w^6-86 v w^7+20 w^8-80 u^7 (v+w)+16 u^6 (11 v^2+13 v w+11 w^2)-8 u^5 (31 v^3+39 v^2 w+39 v w^2+31 w^3)+5 u^4 (61 v^4+4 v^3 w+150 v^2 w^2+4 v w^3+61 w^4)-10 u^3 (29 v^5-23 v^4 w+50 v^3 w^2+50 v^2 w^3-23 v w^4+29 w^5)+u^2 (191 v^6-276 v^5 w+345 v^4 w^2+40 v^3 w^3+345 v^2 w^4-276 v w^5+191 w^6)-2 u (37 v^7-68 v^6 w+66 v^5 w^2+5 v^4 w^3+5 v^3 w^4+66 v^2 w^5-68 v w^6+37 w^7)) z)+b^4 (-3 c^4 (v-w)^2 (12 u^6+4 v^6-10 v^5 w+17 v^4 w^2-10 v^3 w^3+17 v^2 w^4-10 v w^5+4 w^6-36 u^5 (v+w)+u^4 (49 v^2+82 v w+49 w^2)-2 u^3 (19 v^3+41 v^2 w+41 v w^2+19 w^3)+3 u^2 (9 v^4+2 v^3 w+38 v^2 w^2+2 v w^3+9 w^4)-2 u (7 v^5-8 v^4 w+19 v^3 w^2+19 v^2 w^3-8 v w^4+7 w^5)) x^2+2 a^2 c^2 (20 u^8+20 v^8-80 v^7 w+176 v^6 w^2-248 v^5 w^3+305 v^4 w^4-290 v^3 w^5+191 v^2 w^6-74 v w^7+20 w^8-2 u^7 (37 v+43 w)+u^6 (191 v^2+136 v w+233 w^2)-2 u^5 (145 v^3+138 v^2 w+66 v w^2+211 w^3)+5 u^4 (61 v^4+46 v^3 w+69 v^2 w^2-2 v w^3+106 w^4)-2 u^3 (124 v^5-10 v^4 w+250 v^3 w^2-20 v^2 w^3+5 v w^4+211 w^5)+u^2 (176 v^6-312 v^5 w+750 v^4 w^2-500 v^3 w^3+345 v^2 w^4-132 v w^5+233 w^6)+u (-80 v^7+208 v^6 w-312 v^5 w^2+20 v^4 w^3+230 v^3 w^4-276 v^2 w^5+136 v w^6-86 w^7)) x z-3 a^4 (u-v)^2 (4 u^6+4 v^6-14 v^5 w+27 v^4 w^2-38 v^3 w^3+49 v^2 w^4-36 v w^5+12 w^6-2 u^5 (5 v+7 w)+u^4 (17 v^2+16 v w+27 w^2)-2 u^3 (5 v^3+19 v^2 w-3 v w^2+19 w^3)+u^2 (17 v^4-38 v^3 w+114 v^2 w^2-82 v w^3+49 w^4)-2 u (5 v^5-8 v^4 w-3 v^3 w^2+41 v^2 w^3-41 v w^4+18 w^5)) z^2)=0
Let H=X(4)-Orthocenter of ABC. A 1 B 1 C 1 cevian triangle of H. B1 A , C1 A are reflections of B1, C1 on A. L A , line through B1 A , C1...
For any point {a^2 (u - v) (-u + w), b^2 (u - v) (v - w), c^2 (v - w) (-u + w)} on circumcircle of ABC, ellipse has equation:
ReplyDelete-3 a^4 c^4 (u-w)^2 (4 u^6+12 v^6-36 v^5 w+49 v^4 w^2-38 v^3 w^3+27 v^2 w^4-14 v w^5+4 w^6-2 u^5 (7 v+5 w)+u^4 (27 v^2+16 v w+17 w^2)-2 u^3 (19 v^3-3 v^2 w+19 v w^2+5 w^3)+u^2 (49 v^4-82 v^3 w+114 v^2 w^2-38 v w^3+17 w^4)-2 u (18 v^5-41 v^4 w+41 v^3 w^2-3 v^2 w^3-8 v w^4+5 w^5)) y^2+2 a^2 b^2 c^2 y (c^2 (20 u^8+20 v^8-74 v^7 w+191 v^6 w^2-290 v^5 w^3+305 v^4 w^4-248 v^3 w^5+176 v^2 w^6-80 v w^7+20 w^8-2 u^7 (43 v+37 w)+u^6 (233 v^2+136 v w+191 w^2)-2 u^5 (211 v^3+66 v^2 w+138 v w^2+145 w^3)+5 u^4 (106 v^4-2 v^3 w+69 v^2 w^2+46 v w^3+61 w^4)-2 u^3 (211 v^5+5 v^4 w-20 v^3 w^2+250 v^2 w^3-10 v w^4+124 w^5)+u^2 (233 v^6-132 v^5 w+345 v^4 w^2-500 v^3 w^3+750 v^2 w^4-312 v w^5+176 w^6)+u (-86 v^7+136 v^6 w-276 v^5 w^2+230 v^4 w^3+20 v^3 w^4-312 v^2 w^5+208 v w^6-80 w^7)) x+a^2 (20 u^8+20 v^8-86 v^7 w+233 v^6 w^2-422 v^5 w^3+530 v^4 w^4-422 v^3 w^5+233 v^2 w^6-86 v w^7+20 w^8-80 u^7 (v+w)+16 u^6 (11 v^2+13 v w+11 w^2)-8 u^5 (31 v^3+39 v^2 w+39 v w^2+31 w^3)+5 u^4 (61 v^4+4 v^3 w+150 v^2 w^2+4 v w^3+61 w^4)-10 u^3 (29 v^5-23 v^4 w+50 v^3 w^2+50 v^2 w^3-23 v w^4+29 w^5)+u^2 (191 v^6-276 v^5 w+345 v^4 w^2+40 v^3 w^3+345 v^2 w^4-276 v w^5+191 w^6)-2 u (37 v^7-68 v^6 w+66 v^5 w^2+5 v^4 w^3+5 v^3 w^4+66 v^2 w^5-68 v w^6+37 w^7)) z)+b^4 (-3 c^4 (v-w)^2 (12 u^6+4 v^6-10 v^5 w+17 v^4 w^2-10 v^3 w^3+17 v^2 w^4-10 v w^5+4 w^6-36 u^5 (v+w)+u^4 (49 v^2+82 v w+49 w^2)-2 u^3 (19 v^3+41 v^2 w+41 v w^2+19 w^3)+3 u^2 (9 v^4+2 v^3 w+38 v^2 w^2+2 v w^3+9 w^4)-2 u (7 v^5-8 v^4 w+19 v^3 w^2+19 v^2 w^3-8 v w^4+7 w^5)) x^2+2 a^2 c^2 (20 u^8+20 v^8-80 v^7 w+176 v^6 w^2-248 v^5 w^3+305 v^4 w^4-290 v^3 w^5+191 v^2 w^6-74 v w^7+20 w^8-2 u^7 (37 v+43 w)+u^6 (191 v^2+136 v w+233 w^2)-2 u^5 (145 v^3+138 v^2 w+66 v w^2+211 w^3)+5 u^4 (61 v^4+46 v^3 w+69 v^2 w^2-2 v w^3+106 w^4)-2 u^3 (124 v^5-10 v^4 w+250 v^3 w^2-20 v^2 w^3+5 v w^4+211 w^5)+u^2 (176 v^6-312 v^5 w+750 v^4 w^2-500 v^3 w^3+345 v^2 w^4-132 v w^5+233 w^6)+u (-80 v^7+208 v^6 w-312 v^5 w^2+20 v^4 w^3+230 v^3 w^4-276 v^2 w^5+136 v w^6-86 w^7)) x z-3 a^4 (u-v)^2 (4 u^6+4 v^6-14 v^5 w+27 v^4 w^2-38 v^3 w^3+49 v^2 w^4-36 v w^5+12 w^6-2 u^5 (5 v+7 w)+u^4 (17 v^2+16 v w+27 w^2)-2 u^3 (5 v^3+19 v^2 w-3 v w^2+19 w^3)+u^2 (17 v^4-38 v^3 w+114 v^2 w^2-82 v w^3+49 w^4)-2 u (5 v^5-8 v^4 w-3 v^3 w^2+41 v^2 w^3-41 v w^4+18 w^5)) z^2)=0