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Questions of the kind "What's the name for a X that satisfies property Y?"

2 votes
1 answer
122 views

Does this result above six points follow have a name?

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 $CD, EB, AF$ …
2 votes
2 answers
231 views

A necessary and sufficient condition for three diagonals of a hexagon to be concurrent

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 …
5 votes
1 answer
1k views

Is this a new result about hexagon?

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 …
0 votes
1 answer
142 views

Inequality $(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$

Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then $$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$ Equality iff $x=y$ Is the conjecture right? Have you ever seen this in …
2 votes
0 answers
109 views

Another Butterfly theorem — Conway like circle

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 …
3 votes
1 answer
2k views

Does this hexagon theorem have a name?

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, …
2 votes
1 answer
180 views

Is a line associated with antipodal points (the fact, it is the generalization of Simson lin...

First time, I found a line associated with antipodal points, detail: Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ …
1 vote

Is it a new discovery on conic section?

Application of the theorem in post #1. I give a special case and give a proof as follows: Generalization of conjugate of a point: Let $ABC$ be a triangle, and $\Omega$ is arbitrary circumconic of $ABC …
Đào Thanh Oai's user avatar
13 votes
2 answers
2k views

Is it a new discovery on conic section?

I discovered a problem in plane geometry (there are some nice special cases) as follows: Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in …
5 votes
1 answer
426 views

Golden ratio as a property of conic section (is it known?)

I am looking for a proof of a discovery as follows: Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $( …
1 vote
1 answer
352 views

Thirteen-point conic and four-point line, are they new?

We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and …
6 votes
0 answers
319 views

Does this plane geometry theorem have a name (well-known)?

Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points …
11 votes

Does this geometry theorem have a name?

This is theorem 2 (the Parallel tangent theorem) in "Two Applications of the Generalized Ptolemy Theorem" by Shay Gueron.
Alex M.'s user avatar
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