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You can check the conjecture above hold true for $n=1,\cdots ,3000$ in here the geogebra applet Some new equilateral triangles I discovered recently in here: SOME NEW EQUILATERAL TRIANGLES IN A … PLANE GEOMETRY Some Equilateral Triangles Perspective to the Reference Triangle ABC …

His paper Equilateral chordal triangles. …

In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. …

1. Introduction The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 wi …

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by GeoGebra, I found two special points as follow …

Two triangles $A_1B_1C_1$ and $ABC$ are perspective. 3. The triangle $A_1B_1C_1$ and the Morley triangle are homothetic. … Some new equilateral triangles I discovered recently in here: SOME NEW EQUILATERAL TRIANGLES IN A PLANE GEOMETRY Some Equilateral Triangles Perspective to the Reference Triangle ABC …

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems: Let three $(A)$, $(B)$, $ …

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. …

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 …

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 …

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. …

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 …