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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 $CD, EB, AF$ …

When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric rela …

Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent: Three lines $AA', BB', CC'$ are concurrent (let the point o …

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 $A'$, $B'$ …

Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that: Let $ABCD$ be a convex cyclic quadrilateral. …

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^ …

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 …

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?

I proposed a generalization of the Archimedean circle : In this figure $M$ is the midpoint of $AB$, $DE$; $(G)$, $(H)$, $(M)$ are the semicircles. Then two yellow circles are congruent. Question: Is t …

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 …

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 …

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is …

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 …

When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here …