Suppose you have a point X and wish to determine if it is in the Encyclopedia of Triangle Centers. To do so using this page, you need the first normalized trilinear coordinate of X evaluated at (a,b,c) = (6,9,13).
If X is expressed in trilinears as x : y : z, the required number for comparison with the list below is kx, where k is determined as follows:
If X is not on the line at infinity, then k = 2*(area of triangle ABC)/(ax + by + cz). [In this case, kx is simply the directed distance between X and sideline BC of the reference triangle.]If X is on the line at infinity, then k = 1/x + 1/y + 1/z.
If you are starting with barycentric coordinates u : v : w for X, then put x = u/a, y = v/b, z = w/c, and proceed as above.
First coordinate (of normalized trilinears):
triangle centers 1-3053 evaluated at (a,b,c) = (6,9,13)Column 2 shows 1st coordinates for centers as indexed in ETC. Column 4 shows 1st coordinates arranged in increasing order. For example, row 1 indicates that X(1), the incenter, has distance 1.690308509457 from line BC, and that X(348) is the center farthest from BC on its negative side, with distance -468.904101327.
Index Coordinate . . . . . . . Rank Coordinate 1 1.690308509457 348 -468.904101327 2 2.629368792488 1105 -309.394308647 3 6.782362894196 615 -228.134102794 4 -5.67661941092 2947 -169.079821500 5 0.552871741634 1786 -166.978966131 6 0.992908495065 2960 -165.912433836 7 0.744160979006 2191 -165.650233926 8 4.507489358552 1904 -161.920265180 9 3.571972699229 526 -137.215789466 10 3.098898934004 2806 -106.835721957 11 2.643135121245 1831 -104.620852749 12 1.334163924620 1578 -99.6666422158 13 0.075945151804 2978 -91.9990135745 14 4.373431772013 95 -89.7364361063 15 3.102444020654 1810 -75.9145184033 16 -6.78421149583 3004 -71.7442056236 17 1.386654550069 2537 -67.9794087315 18 -16.2806739977 2200 -67.0055484832 19 -1.48051239778 2915 -65.6111228848 20 19.24134519931 1677 -61.9574298147 21 3.177166860314 2950 -61.1314287408 22 -24.3634496369 2592 -58.7364798515 23 -3.05356651787 2431 -58.6948006119 24 1.591165914392 2651 -58.3473232727 25 -.546404736965 47 -56.5124887342 26 -2.11677603447 710 -55.5557617832 27 -2.59682603618 1850 -54.1036844244 28 -.868703140076 84 -52.7703752224 29 5.278261169382 1021 -52.5681623401 30 1.134604388331 722 -50.9503190095 31 0.542275931741 661 -49.5823829440 32 0.280712446185 732 -48.7637393355 33 5.246983589481 740 -48.1329977116 34 -.119171170782 913 -46.9708108001 35 3.284689530119 2654 -45.3454659344 36 -2.57528678665 521 -44.4753160817 37 2.090823377159 2571 -43.6596673834 38 2.431598760008 2584 -43.5788297110 39 1.564479646462 2526 -41.0503495153 40 11.87441727893 494 -40.6395857094 41 1.800613978898 2499 -40.4903531268 42 1.283884145284 2041 -39.7151920791 43 0.257220860134 1332 -39.7115198584 44 -6.39576192767 770 -36.7350850360 45 2.533082047975 2394 -36.6700161366 46 -1.31607046308 533 -36.4162059254 47 -56.5124887342 1593 -34.7490389016 48 16.29513001394 2396 -34.6386895083 49 1.301655102607 2436 -31.6824610836 50 3.400616763711 1491 -29.5401534747 51 0.120625582707 2787 -29.1220237257 52 -.743866735147 1161 -28.2024585343 53 -.134512425199 945 -28.1073063546 54 -6.18775905246 2631 -26.9004284840 55 2.635469416137 2438 -26.0383006582 56 0.187119023187 1826 -25.0039153173 57 0.380455291517 2512 -24.8149151002 58 0.751932245361 1276 -24.7574798721 59 0.114418313907 22 -24.3634496369 60 1.901815483389 1952 -24.2560400024 61 1.921708280859 2927 -23.5225478069 62 -.374701511312 2804 -23.0294612605 63 5.874900573472 2578 -22.4709026507 64 31.62732406603 2473 -22.0398251043 65 0.529726313505 1437 -21.8796218442 66 9.640588374560 2156 -21.7984073221 67 5.987634908298 2261 -21.0896149717 68 1.370063903568 1158 -20.4479789717 69 5.902289387334 2075 -20.3757449380 70 5.721069014471 1595 -20.2128291562 71 5.850835756503 2530 -20.1227203506 72 5.668071554504 822 -19.4524781772 73 -2.30612393141 195 -19.1578809991 74 12.65716281429 151 -18.9194928849 75 3.706459623146 1734 -18.9153571296 76 4.759147084540 2637 -18.3964543323 77 -4.33130712823 525 -18.1498675002 78 5.339129461605 382 -18.1356017160 79 0.046914305978 1498 -18.0625982776 80 3.595961733033 1536 -18.0362829908 81 1.333662438745 650 -18.0299574342 82 1.057543146403 469 -17.5945797685 83 1.810750169531 146 -17.4262192511 84 -52.7703752224 286 -17.3901070230 85 1.369706806416 879 -17.3804195730 86 2.176292659804 239 -17.2685572047 87 5.357195925404 2962 -17.2485702444 88 0.562219245046 1623 -16.9939027901 89 1.010174849101 1345 -16.9347947868 90 2.722260516137 383 -16.8342626787 91 3.738569256341 152 -16.6003317266 92 31.41053152638 708 -16.5062951936 93 18.57341790506 796 -16.4876177065 94 -.265365586503 657 -16.3290180536 95 -89.7364361063 18 -16.2806739977 96 -1.29974999170 962 -15.8607281804 97 -13.3395865508 2914 -15.6102199858 98 10.12599633474 2313 -15.4797928281 99 3.438729453648 1606 -15.4718768636 100 2.601836134975 2624 -15.4496935101 101 1.320506736351 680 -15.1953171805 102 13.40379963121 2389 -15.0408095801 103 12.24421905204 1750 -14.9385770214 104 10.96288965341 652 -14.8020878168 105 2.693929186947 2532 -14.7829136452 106 0.410148388324 2900 -14.7366664852 107 0.245357987935 2400 -14.5814098173 108 -.081760732884 1763 -14.4186262751 109 0.160926157179 1885 -14.3980384899 110 0.907562974101 2590 -14.3712956877 111 1.207441876625 2774 -14.2900910367 112 -.200462721488 2657 -14.2444229081 113 -2.38452821841 153 -14.0376729293 114 -1.11894497863 720 -13.9933413299 115 2.224688461909 198 -13.6496893798 116 3.283799820557 1526 -13.6224953644 117 -2.75784662687 656 -13.5339667729 118 -2.17805633728 2167 -13.4914250946 119 -1.53739163797 97 -13.3395865508 120 2.597088595259 2212 -13.3250317690 121 3.738978994571 2522 -13.3232420266 122 3.821374194765 566 -13.3179491972 123 3.984933555175 1563 -13.2965177521 124 3.863590110143 1337 -13.2093170038 125 3.490271701682 2224 -13.1872946875 126 3.340332250420 1512 -13.0278389668 127 4.044284549477 2136 -12.8463446718 128 4.231492060174 2500 -12.7938868215 129 3.956071987549 2132 -12.7274488766 130 -2.85032850427 1941 -12.6783314099 131 -3.11386215833 1493 -12.6728200258 132 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-10.4616180742 158 0.730217888460 1545 -10.4015886651 159 -2.74539049216 1628 -10.3985619592 160 -5.17858468346 430 -10.3565043918 161 6.599863926710 1152 -10.3431095583 162 -.434609584744 2799 -10.2446951537 163 0.422785826392 900 -10.1976911976 164 5.601337607766 2516 -10.1418510567 165 8.479714355776 1867 -10.0893341111 166 4.147361892876 2523 -10.0261601712 167 14.23646785179 2348 -10.0118273252 168 10.64589058722 2939 -9.97115373114 169 1.504667690499 2290 -9.95967948552 170 15.53510923631 1910 -9.94768477776 171 1.379817388436 2577 -9.85188843874 172 0.774827726345 706 -9.84320283086 173 0.865451887563 1538 -9.82864249480 174 1.175132454770 2227 -9.80099271520 175 -7.15162777590 2103 -9.79303344768 176 1.190492903652 2614 -9.79156417632 177 0.975427635249 1535 -9.79070939113 178 2.359244703059 2430 -9.75601284437 179 1.257739557595 1513 -9.74219161042 180 1.184017677849 909 -9.63439255159 181 0.373069446401 2159 -9.63118544066 182 3.887635694631 1558 -9.62311207155 183 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