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The straight line on which all points at infinity lie. The line at infinity is central line (Kimberling 1998, p. 150), and has trilinear equation

which follows from the fact that a real triangle will have positive area, and therefore that

The line at infinity passes through Kimberling centers for (the Euler infinity point),

511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 674, 680, 688, 690, 696, 698, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724, 726, 730, 732, 734, 736, 740, 742, 744, 746, 752, 754, 758, 760, 766, 768, 772, 776, 778, 780, 782, 784, 786, 788, 790, 792, 794, 796, 802, 804, 806, 808, 812, 814, 816, 818, 824, 826, 830, 832, 834, 838, 888, 891, 900, 912, 916, 918, 924, 926, 928, 952, 971, 1154, 1499, 1503, 1510, 1912, 1938, 1946, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 2393, 2574, 2575, 2771, 2772, 2773, 2774, 2775, 2776, 2777, 2778, 2779, 2780, 2781, 2782, 2783, 2784, 2785, 2786, 2787, 2788, 2789, 2790, 2791, 2792, 2793, 2794, 2795, 2796, 2797, 2798, 2799, 2800, 2801, 2802, 2803, 2804, 2805, 2806, 2807, 2808, 2809, 2810, 2811, 2812, 2813, 2814, 2815, 2816, 2817, 2818, 2819, 2820, 2821, 2822, 2823, 2824, 2825, 2826, 2827, 2828, 2829, 2830, 2831, 2832, 2833, 2834, 2835, 2836, 2837, 2838, 2839, 2840, 2841, 2842, 2843, 2844, 2845, 2846, 2847, 2848, 2849, 2850, 2851, 2852, 2853, 2854, 2869, 2870, 2871, 2872, 2873, 2874, 2875, 2876, 2877, 2878, 2879, 2880, 2881, and 2882.

The line at infinity satisfies the remarkable property of being its own complement, and therefore also its own anticomplement.

Instead of the three reflected segments concurring for the isogonal conjugate of a point on the circumcircle of a triangle, they become parallel (and can be considered to meet at infinity). As varies around the circumcircle, varies through a line called the line at infinity. Every line is perpendicular to the line at infinity.

Poncelet was the first to systematically employ the line at infinity (Graustein 1930).

See also

Circular Point at Infinity, Point at Infinity

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., 1888.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 30, 1930.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. §10 in An Elementary Treatise on Modern Pure Geometry. London, England: Macmillian, p. 6, 1893.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 141-142, 1991.

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Line at Infinity

Cite this as:

Weisstein, Eric W. "Line at Infinity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LineatInfinity.html

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