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Does this result above six points follow have a name? Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ …

In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle …

There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera. Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at …

Theorem 9.1 in this paper as follows is a generalization of Turker circle. Turker circles is a generalization of many circles as: Cosine Circle, circum circle, First Lemoine Circle, Gallatly Circle, K …

Have You seen these result as follows before? In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral. In …

More than seven years ago. I posted this problem in stackexchange: Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$ …

Locus equation of the point $O_1$, In Cartesian coordinates, as follows: $$x=\frac{1}{2}t\frac{t^2-3}{t^2-1}$$ $$y=\frac{1}{2}\frac{t^2+1}{1-t^2}$$ where $-1<t<1$ or the equation: $$x^2-y^2=\frac{2y^ …

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 …

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?

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 …

Question : Do you know this property of a hexagon? Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, …

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to: \begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\s …

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 …

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and p …

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure below. I am looking for a proof that: $$DE+EF+FD \le (DG+DH+E …