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 1-3514 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(3207) is the center farthest from BC (except for points on the infinity line) on its negative side, with distance -678.377148462.
Index Coordinate . . . . . . . Rank Coordinate 1 1.690308509457 3207 -678.377148462 2 2.629368792488 348 -468.904101327 3 6.782362894196 1105 -309.394308647 4 -5.67661941092 615 -228.134102794 5 0.552871741634 3191 -221.167529104 6 0.992908495065 3355 -169.876745382 7 0.744160979006 2947 -169.079821500 8 4.507489358552 1786 -166.978966131 9 3.571972699229 2960 -165.912433836 10 3.098898934004 2191 -165.650233926 11 2.643135121245 1904 -161.920265180 12 1.334163924620 526 -137.215789466 13 0.075945151804 3054 -113.162937858 14 4.373431772013 2806 -106.835721957 15 3.102444020654 1831 -104.620852749 16 -6.78421149583 1578 -99.6666422158 17 1.386654550069 3429 -99.2268703479 18 -16.2806739977 2978 -91.9990135745 19 -1.48051239778 95 -89.7364361063 20 19.24134519931 3424 -78.5522836775 21 3.177166860314 1810 -75.9145184033 22 -24.3634496369 3004 -71.7442056236 23 -3.05356651787 2537 -67.9794087315 24 1.591165914392 2200 -67.0055484832 25 -.546404736965 2915 -65.6111228848 26 -2.11677603447 1677 -61.9574298147 27 -2.59682603618 2950 -61.1314287408 28 -.868703140076 3198 -60.0591266419 29 5.278261169382 2592 -58.7364798515 30 1.134604388331 2431 -58.6948006119 31 0.542275931741 2651 -58.3473232727 32 0.280712446185 47 -56.5124887342 33 5.246983589481 710 -55.5557617832 34 -.119171170782 1850 -54.1036844244 35 3.284689530119 84 -52.7703752224 36 -2.57528678665 1021 -52.5681623401 37 2.090823377159 722 -50.9503190095 38 2.431598760008 661 -49.5823829440 39 1.564479646462 732 -48.7637393355 40 11.87441727893 740 -48.1329977116 41 1.800613978898 913 -46.9708108001 42 1.283884145284 2654 -45.3454659344 43 0.257220860134 521 -44.4753160817 44 -6.39576192767 3467 -43.9756210582 45 2.533082047975 2571 -43.6596673834 46 -1.31607046308 2584 -43.5788297110 47 -56.5124887342 3405 -42.8991037814 48 16.29513001394 2526 -41.0503495153 49 1.301655102607 494 -40.6395857094 50 3.400616763711 2499 -40.4903531268 51 0.120625582707 2041 -39.7151920791 52 -.743866735147 1332 -39.7115198584 53 -.134512425199 3426 -39.1421903070 54 -6.18775905246 770 -36.7350850360 55 2.635469416137 2394 -36.6700161366 56 0.187119023187 533 -36.4162059254 57 0.380455291517 3471 -35.1766327018 58 0.751932245361 1593 -34.7490389016 59 0.114418313907 2396 -34.6386895083 60 1.901815483389 2436 -31.6824610836 61 1.921708280859 3384 -30.9878290276 62 -.374701511312 3427 -30.8900097601 63 5.874900573472 3146 -30.5945840211 64 31.62732406603 3307 -30.1699447611 65 0.529726313505 1491 -29.5401534747 66 9.640588374560 3283 -29.1244055198 67 5.987634908298 2787 -29.1220237257 68 1.370063903568 3223 -29.0851428944 69 5.902289387334 3446 -28.7176310154 70 5.721069014471 1161 -28.2024585343 71 5.850835756503 945 -28.1073063546 72 5.668071554504 2631 -26.9004284840 73 -2.30612393141 2438 -26.0383006582 74 12.65716281429 3185 -25.5049439040 75 3.706459623146 1826 -25.0039153173 76 4.759147084540 2512 -24.8149151002 77 -4.33130712823 1276 -24.7574798721 78 5.339129461605 3358 -24.5992012616 79 0.046914305978 22 -24.3634496369 80 3.595961733033 1952 -24.2560400024 81 1.333662438745 2927 -23.5225478069 82 1.057543146403 2804 -23.0294612605 83 1.810750169531 2578 -22.4709026507 84 -52.7703752224 2473 -22.0398251043 85 1.369706806416 3372 -22.0253432065 86 2.176292659804 1437 -21.8796218442 87 5.357195925404 2156 -21.7984073221 88 0.562219245046 3455 -21.3285072790 89 1.010174849101 2261 -21.0896149717 90 2.722260516137 3509 -21.0442660840 91 3.738569256341 1158 -20.4479789717 92 31.41053152638 2075 -20.3757449380 93 18.57341790506 1595 -20.2128291562 94 -.265365586503 2530 -20.1227203506 95 -89.7364361063 3250 -19.9307059588 96 -1.29974999170 822 -19.4524781772 97 -13.3395865508 3287 -19.2907936184 98 10.12599633474 195 -19.1578809991 99 3.438729453648 151 -18.9194928849 100 2.601836134975 1734 -18.9153571296 101 1.320506736351 2637 -18.3964543323 102 13.40379963121 525 -18.1498675002 103 12.24421905204 382 -18.1356017160 104 10.96288965341 1498 -18.0625982776 105 2.693929186947 1536 -18.0362829908 106 0.410148388324 650 -18.0299574342 107 0.245357987935 469 -17.5945797685 108 -.081760732884 146 -17.4262192511 109 0.160926157179 286 -17.3901070230 110 0.907562974101 879 -17.3804195730 111 1.207441876625 239 -17.2685572047 112 -.200462721488 2962 -17.2485702444 113 -2.38452821841 1623 -16.9939027901 114 -1.11894497863 1345 -16.9347947868 115 2.224688461909 383 -16.8342626787 116 3.283799820557 152 -16.6003317266 117 -2.75784662687 708 -16.5062951936 118 -2.17805633728 796 -16.4876177065 119 -1.53739163797 657 -16.3290180536 120 2.597088595259 18 -16.2806739977 121 3.738978994571 962 -15.8607281804 122 3.821374194765 3189 -15.6635255209 123 3.984933555175 2914 -15.6102199858 124 3.863590110143 2313 -15.4797928281 125 3.490271701682 1606 -15.4718768636 126 3.340332250420 2624 -15.4496935101 127 4.044284549477 680 -15.1953171805 128 4.231492060174 2389 -15.0408095801 129 3.956071987549 1750 -14.9385770214 130 -2.85032850427 652 -14.8020878168 131 -3.11386215833 2532 -14.7829136452 132 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