Cubics Associated with Triangles of Equal Areas
نویسندگان
چکیده
The locus of a point X for which the cevian triangle of X and that of its isogonal conjugate have equal areas is a cubic that passes through the 1st and 2nd Brocard points. Generalizing from isogonal conjugate to P -isoconjugate yields a cubic Z(U,P ) passing through U ; if X is on Z(U, P ) then the P isoconjugate of X is on Z(U,P ) and this point is collinear with X and U . A generalized equal areas cubic Γ(P ) is presented as a special case of Z(U,P ). If σ = area( ABC), then the locus of X whose cevian triangle has prescribed oriented area Kσ is a cubic Λ(P ), and P is determined if K has a certain form. Various points are proved to lie on Λ(P ).
منابع مشابه
On Cyclical Quabtic Surfaces in Space of N Dimensions
Professor White's paper was a further development of the topic considered in the paper presented by him at the Columbus meeting of the Society. Each mixed concomitant (2, 2) of the cubic defines (as in the paper referred to) two covariant nets of conies. These are polars of two cubics of the syzygetic sheaf ; the totality of such is exactly that entire sheaf of cubics. But these concomitants (2...
متن کاملTilings of the plane with unit area triangles of bounded diameter
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.
متن کاملLeast - Perimeter Partitions of the Sphere
We consider generalizations of the honeycomb problem to the sphere S and seek the perimeter-minimizing partition into n regions of equal area. We provide a new proof of Masters’ result that three great semicircles meeting at the poles at 120 degrees minimize perimeter among partitions into three equal areas. We also treat the case of four equal areas, and we prove under various hypotheses that ...
متن کاملThe Isosceles Trapezoid and its Dissecting Similar Triangles
Isosceles trapezoids are dissected into three similar triangles and rearranged to form two additional isosceles trapezoids. Moreover, triangle centers, one from each similar triangle, form the vertices of a centric triangle which has special properties. For example, the centroidal triangles are congruent to each other and have an area one-ninth of the area of the trapezoids; whereas, the circum...
متن کاملOn the Area Discrepancy of Triangulations of Squares and Trapezoids
In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid T (α) whose vertices have the coordinates (0, 0), (0, 1), (1, 0), and (α, 1) cannot be triangulated into any number of triangles of equal areas if α > 0 is transcendental. In this paper we first establish a new a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002