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Search options not deleted user 122662

This tag is used if a reference is needed in a paper or textbook on a specific result.

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 …
Đào Thanh Oai's user avatar
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 …
Đào Thanh Oai's user avatar
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 …
Đào Thanh Oai's user avatar
4 votes
0 answers
334 views

The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$

A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted $g_n$ or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my comp …
Đào Thanh Oai's user avatar
2 votes
0 answers
211 views

A generalization of the Archimedean circle

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 …
Đào Thanh Oai's user avatar
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, …
Đào Thanh Oai's user avatar
11 votes
3 answers
704 views

An open triangle problem in plane geometry

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 …
Đào Thanh Oai's user avatar
4 votes
0 answers
372 views

Two triangles have the same centroid theorem

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 …
Đào Thanh Oai's user avatar
1 vote
0 answers
172 views

Four incenters lie on a circle-Does this theorem have a name?

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 …
Đào Thanh Oai's user avatar
2 votes
0 answers
148 views

Does this theorem on tangential quadrilateral have a name?

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 …
Đào Thanh Oai's user avatar
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 $P'$ …
Đào Thanh Oai's user avatar
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 …
Đào Thanh Oai's user avatar
5 votes
1 answer
357 views

Discovered 240 new circles associated with Pascal's line

I am looking for a proof or a reference request for a problem as follows: Problem: Let a cyclic hexagon with sidelines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$ and $l_1 \cap l_4 =A$, $l_3 \cap l_6 = …
Đào Thanh Oai's user avatar
2 votes
1 answer
97 views

There is no general method to construct n-regular polygon such that the given n-polygon insc...

Conjecture 1: With $n\ge 5$, given general n-polygon, there is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon (with one and only one ver …
Đào Thanh Oai's user avatar
5 votes
1 answer
266 views

Is it a known property of positive integers $n> 2 $ that one must have $n < \mathrm{rad}(n(n...

Let $P(n)$ be the statement that $$n < \mathrm{rad}(n(n-1)(n-2)),$$ where $\mathrm{rad}$ is the radical of an integer, that is defined as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ pri …
Đào Thanh Oai's user avatar

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