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Conic Construction of a Triangle from the Feet of Its Research Angle bisectors

Conic Construction of a Triangle from the Feet of Its Research Angle bisectors be content with: The angle bisectors problem, the cubic Ka, Existence of solutions to the angle bisectors problem, the hyperbola Ca, conic solution of the angle bisectors problem, examples, the angle bisectors problem for a right triangle, triangles from the feet of external angle bisectors. Invite you to consult the documentation | Conic Construction of a Triangle from the Feet of Its Angle Bisectors Paul Yiu Abstract. We study an extension of the problem of construction of a triangle from the feet of its internal angle bisectors. Given a triangle ABC, we give a conic construction of points which are the incenter or excenters of their own anticevian triangles with respect to ABC. If the given triangle contains a right angle, a very simple ruler-and-compass construction is possible. We also examine the case when the feet of the three external angle bisectors are three given points on a line. 1. The angle bisectors problem In this note we address the problem of construction of a triangle from the endpoints of its angle bisectors. This is Problem 138 in Wernick’s list [3]. The corresponding problem of determining a triangle from the lengths of its angle bisectors have been settled by Mironescu and Panaitopol [2]. A′ B C P B′ C′ A Figure 1. The angle bisectors problem Given a triangle ABC, we seek, more generally, a triangle A′ B ′ C ′ such that the lines A′ A, B ′ B, C ′ C bisect the angles B ′ A′ C ′ , C ′ A′ B ′ , A′ C ′ B ′ , internally or externally. In this note, we refer to this as the angle bisectors problem. With reference to triangle ABC, A′ B ′ C ′ is the anticevian triangle of a point P , which is the incenter or an excenter of triangle A′ B ′ C ′ . It is an excenter if two of the lines A′ P , B ′ P , C ′ P are external angle bisectors and the remaining one an internal angle bisector. For a nondegenerate triangle ABC, we show in §3 that the angle bisectors problem always have real solutions, as intersections of three cubics. We proceed to provide a conic solution in §§4, 5, 6. The particular case of right triangles has an To appear in Journal for Geometry and Graphics, 12 (2008) 133–144. 1 2 P. Yiu elegant ruler-and-compass solution which we provide in §7. Finally, the construction of a triangle from the feet of its external angle bisectors will be considered in §8. In .