Given $ΔABC$ and angles $α,β,γ\in(0,π)$ with $α+β+γ=π$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=α$, $\aangle E=β$, $\aangle F=γ$, (iii) $ΔDEF$ has the minimal area in the class of triangles satisfying (i) and (ii). In particular, we show that minimizer $ΔDEF$, exists, is unique and is a pedal triangle, corresponding to a certain pedal point $P$. Permuting the roles played by the angles $α,β,γ$ 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.
Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the…
An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circles
Kira Adaricheva和Madina Bolat最近证明了，如果$U_0$和$U_1$是三角形中的圆，顶点为$A_0,A_1,A_2$，则存在$j\in \{0,1,2\}$和$k\in\{0,1\}$，使得$U_{1-k}$包含在$U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$的凸包中。我们给这个结果一个简短的新的证明，我们指出，一个简单的推广shperes失败。
In order to state the theorem in the title formally and to review its rigorous proof, we extend and make more precise the Uspenskiy-Shen-Akopyan-Fedorov model of Euclidean constructions with arbitrary points; we also introduce formalizations for infinite configurations and for the projective plane. We exemplify the proof method by simpler and not so well known results that it is impossible to construct the unit length, or a given point, by compass and straightedge from nothing by means of classical arbitrary points. On the other hand we construct any given point by compass and straightedge fro…
We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple …
