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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 …
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.