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I am looking for a proof that: if $A_{11}A_{12}...A_{1n}$; $A_{21}A_{22}...A_{2n}$; $\cdots$; $A_{i1}A_{i2}...A_{in}$; $\cdots$; $A_{m1}A_{m2}...A_{mn}$ are $m$ oriented regular polygons ($n$-gons), w …

Please don't close this question. Because there is simple configuration with 57 vote up, and don't close. Why you vote up that question and You vote to close this question? A problem I posed at here …

Let $ABCD$ be a quadrilateral, $P$ be a point in the plane let $E$, $F$ be the projections of the incenters of triangles $\triangle CPB$, $\triangle BPA$ onto $PB$ respectively; Let $G$, $H$ be the …

The result as follows from special configuration of merge Nine Circle Theorem and Eight Circle theorem but it is new: Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in th …

I am looking for a proof of a generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem and van Aubel theorem, and Finsler–Hadwiger theorem in one configuration, as follows: Let four po …

Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, …

Given $2n+1$ fixed points: $A_1, A_2,....,A_{2n+1}$ and point $P$. Let $B_1$ is the reflection of $P$ in $A_1$, $B_2$ is the reflection of $B_1$ in $A_2$,...., $B_{2n+1}$ is the reflection of $B_{2n}$ …

The solution of problem in our paper On an Extension of Miquel's Theorem to a Cyclic Hexagon; Relative configuration in here On the eight circles theorem and its dual

I am looking for a proof of the problem as follows: Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$ …

An equilateral triangle constructed from a reference triangle is a topic which is intersested by plane geometry lovers. See Napoleon equilateral triangle, Morley equilateral triangle....In this topic …

Ono's inequality is true for acute triangle but false with general triangles. The inequality as follows is false with general triangls but I think it true with acute triangle (follows answer by Fedor …

I am looking for a proof or a reference request for a problem as follows: Problem: Let a cyclic hexagon with sidelines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$ and $l_1 \cap l_4 =A$, $l_3 \cap l_6 = …

Abstract: In the figure belows: Three lines through center of pair opposite red circle are concurrent. This is a statement of Dao's theorem on six circumcenter, a new theorem in plane geometry which w …

In this configuration as follows, we have a nice formula: $$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$ Is the formula known? and can we generalized for higher dimensions?

Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2 …