Triangles with Vertices Equidistant to a Pedal Triangle

2017

Two very different, yet related, triangle constructions are examined, based on a given reference triangle and on a triple of signed angles. Th e produce triangles that are in perspective with the reference triangle and with e ach other, using the same center of perspective. The first construction is r ather wellknown, and produces a Kiepert-Morley-Hofstadter-Kimberling triangle . A new circumconic is associated with this construction. The second construction g eneralizes work of D. M. Bailey and J. Van Yzeren. A number of known ce tral triangles are obtainable using one or both of these constructions.

arXiv (Cornell University), 2023

We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find a nice relation involving the radii of the inner and outer Apollonius circles of the three circles in the triad.

2010

We give a new proof of the formula expressing the area of the triangle whose vertices are the projections of an arbitrary point in the plane onto the sides of a given triangle, in terms of the geometry of the given triangle and the location of the projection point. Other related geometrical constructions and formulas are also presented.

2011

The r-Hofstadter triangle and the (1 − r)-Hofstadter triangle are proved perspective, and homogeneous trilinear coordinates are found for the perspector. More generally, given a triangle DEF inscribed in a reference triangle ABC, triangles A ′ B ′ C ′ and A ′′ B ′′ C ′′ derived in a certain manner from DEF are perspective to each other and to ABC. Trilinears for the three perspectors, denoted by P * , P1, P2 are found (Theorem 1) and used to prove that these three points are collinear. Special cases include (Theorems 4 and 5) this: if X and X ′ are an antipodal pair on the circumcircle, then the perspector P * = X ⊕ X ′ , where ⊕ denotes crosssum, is on the nine-point circle. Taking X to be successively the vertices of a triangle DEF inscribed in the circumcircle thus yields a triangle D ′ E ′ F ′ inscribed in the nine-point circle. For example, if DEF is the circumtangential triangle, then D ′ E ′ F ′ is an equilateral triangle.

2017

The paper presents groups of triangles inscribed in a given triangle ABC that have the same Brocard angle as 4ABC . Some of them are similar to 4ABC and some are not. The central group consists of the exterior and interior Pappus triangles of 4ABC . We prove that the Miquel points of Pappus triangles produce pedal triangles that have the same Brocard angle as 4ABC . Furthermore, we prove that the locus of all points, that produce pedal triangles inside 4ABC with the same Brocard angle, is a circle. For Miquel points of exterior Pappus triangles, we prove that all these points are located on the Brocard circle.

Learning and Teaching Mathematics, 2018

Starting with an elementary result for the median triangle of a triangle in plane geometry, this result is generalized by asking 'what-if' questions. Ith therefor gives an example of problem posing, experimentation, conjecturing and proof. The theorems of Ceva, Desargues and Pascal are used to prove the observed conjectures.

2020

A Circumconic passes through a triangle's vertices. We define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a 3-periodic. We study its properties and associated loci.

2016

If ABC is a given triangle in the plane, P is any point not on the extended sides of ABC or its anticomplementary triangle, Q is the complement of the isotomic conjugate of P with respect to ABC, DEF is the cevian triangle of P , and D0 and A0 are the midpoints of segments BC and EF , respectively, a synthetic proof is given for the fact that the complete quadrilateral defined by the lines AP,AQ,D0Q,D0A0 is perspective by a Euclidean half-turn to the similarly defined complete quadrilateral for the isotomic conjugate P ′ of P . This fact is used to define and prove the existence of a generalized circumcenter and generalized orthocenter for any such point P .

2010

Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$, $\aangle F=\gamma$, (iii) $\Delta DEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $\Delta DEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $\alpha,\beta,\gamma$ in (ii), yields a total of six such area-minimizing triangles, which are pedal relative to six pedal points, say, $P_1,....,P_6$. The main result of the paper is the fact that there exists a circle which contains all six points.

2022

In this paper we have proven 6 new theorems about inscribed triangle in a circle. If we extend the three medians of any inscribed triangle in a circle, the extended medians will intersect the circumference of the circle at three points. Connecting those three points we will get a new triangle. The main purpose of this paper is to describe the properties of the second triangle obtained by extending the medians of the first triangle.