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Questions tagged [triangle-centres]

A triangle centre is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure.

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Identifying the centroid, incenter, circumcenter, and orthocenter of a triangle from a set of vector equations

Now the question is: Let $P_i$, with $i=1,2,3,4$, be points lying in plane of $\triangle A_1 A_2 A_3$, satisfying $$\begin{align} &\;\overrightarrow{P_2 A_1} \cdot \overrightarrow{P_2 A_2} + \...
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How to calculate the radius of an outer circle from two symmetric internally tangent circles? (Applied Wind Turbine Problem 🔧)

Motivation & Context The below problem is related to an open-source CAD model of a locally-manufactured small wind turbine. The generator of the turbine is made from two rotating magnetic disks ...
gbroques's user avatar
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Center of outer Soddy circle lies on the same side of lines $AC,BC$ where $AB$ is the longest side

Let $AB$ be the longest side of $\triangle ABC$. I want to prove the center of outer Soddy circle either lies on the intersection of the inner sides of lines $AC,BC$ or lie on the intersection of the ...
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Distance between triangle incenter and vertices

after many researches on the subject, I can't find any convincing argument anywhere, so I come to you about this problem which has been brought by some of my high school students. Let $ABC$ be a ...
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what is the trilinear coordinates of the intersection between these 2 circles

(I really searched on the internet, I found NOTHING) In a triangle, we can draw the 9-point Euler circle, and the 3 excircles of that triangle. there always is an intersection point between the Euler ...
Pierre Carlier's user avatar
6 votes
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How would you prove that the orthocentres of triangles with one moving point form a circular arc?

Recently, I was working on some olympiad geometry problems, and I noticed that if two points $A,B$ are fixed on a circle $\omega$, then a variable point $C$ on $\omega$ defines a set of orthocentres ...
TheMathsPerson's user avatar
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Locus of centroid, given the slopes of the sides

Find the locus of the centroid of a triangle if it is known that its orthocentre is at the origin and the slopes of the sides of the triangle are $ m_1, m_2, m_3$ respectively. I solved this problem ...
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How to find the side lengths of a triangle in the incircle of a Pythagorean triple

Given a Pythagorean triple with an incircle, how do I find the sides of a triangle connecting the triple's tangents to that incircle. In the diagram below, I know how to find side $\,i\,$ but not the ...
poetasis's user avatar
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Given a point $P$ in regular tetrahedron $T$, is there a triangle with vertices on the boundary of $T$ whose center of mass is $P$?

Here's a "simple'' problem in basic geometry. Given is a regular tetrahedron $T$. Consider the Barycentric coordinate system on $T$. Given is also a point $P\in T$ in the interior of $T$ such ...
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Generalisation of the Inner Soddy Center

Would it theoretically be possible to generalise the Inner Soddy Center (ISC) to more than three points? I have strong reasons to believe it is possible, but not the ability to do it myself, though I'...
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Concyclic points determined by a triangle

Let $P$ be the orthocenter of $\triangle ABC$, and let $A'$ be the reflection of $A$ about $\overline{BC}$. Show that $A'$, $B$, $C$, and $P$ are concyclic (i.e. lie on the same circle). Maybe this ...
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Is this a new point on the nine-point-circle of a triangle?

I was trying to get a feel for how to solve another question about the largest triangle that can fit in a unit square, by constructing the smallest enclosing square of a triangle in Geogebra. While ...
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Any ideas...Prove that the centres of the circles circumscribing triangles $EBA$, $FBD$, and $GCA$ form an equilateral triangle...

question Consider the rhombus $ABCD$ with centre $O$, with $\angle DAB < 60°$ and the equilateral triangle $ABE$ so that the points $E$ and $D$ are on either side of the line $AB$. The center ...
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A "New" Special Point in a Triangle.

I was playing with the software Geometry Expressions and I was exploring generalizations of special points in triangles (centroid, orthocenters, etc.) when I stumbled upon this construction. J is ...
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The horizon of the heptagonal triangle

I have discovered this interesting property of the heptagonal triangle while making constructions in GeoGebra. Let $\triangle_0$ be a heptagonal triangle with the orthocenter $H_0$, $\triangle_1$ the ...
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Distances of Fermat point from vertices of a triangle

Consider a triangle $ABC$ and a point $T$. Given that $∠ATB=∠ATC=∠BTC=120°$ and $AC=3$, $BC=4$, $∠ACB=90°$, find $(9BT + 7CT)/AT$. From the question we can imply that the point $T$ is the Fermat-...
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Two parallel asymptotes of a triangle

This is a development on my previous question Displacement of the excentral triangles My idea is that a converging sequence of constructions may not be related to the known triangle centers, but ...
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Displacement of the excentral triangles

The definition from the Wolfram MathWorld page: Excentral Triangle Let $\triangle_1$ be the excentral triangle of a triangle $\triangle_0$, $\triangle_2$ the excentral triangle of $\triangle_1$, etc. ...
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Circumellipse of elongated parallelogram

I am looking for the parameters/equation of an ellipse that would pass through all the vertices of an elongated parallelogram. The ellispe has 5 free parameters: center $(x_0,y_0)$, major and minor ...
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Does the sequence of circumcircle mid-arc triangles converge to an equilateral triangle?

This is a follow-up question on The triangle of intersections of angle bisectors and perpendicular bisectors of a triangle Let $\triangle_1$ be a triangle, $\triangle_2$ the circumcircle mid-arc ...
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Equal angles in the circumcircle mid-arc triangle

I have "discovered" this property while playing with the circumscribed circle of a triangle in GeoGebra. Let $ABC$ be a triangle, $A'$, $B'$, $C'$ the intersections of the corresponding ...
Alex C's user avatar
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5 votes
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A special point of a triangle related to the nine point circle

Playing with the nine point circle in GeoGebra, I have "discovered" a special point. I assume it is a well known point, but I cannot find its name and properties. Description: Let $ABC$ be a ...
Alex C's user avatar
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3 votes
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What is the relation between a triangle’s centroid and its pedal triangle?

The pedal triangle of a point inside a triangle is the triangle formed by connecting the three feet of the perpendiculars drawn from that point to each side of the triangle. What is the relation ...
Alex Wang's user avatar
2 votes
2 answers
220 views

A formula involving the projection of the orthocenter onto a median

I found out this problem when I was trying to solve another geometry problem but I stuck to solve it. Please help me. Thanks Let $ABC$ be an acute triangle with $M$ be the midpoint of $BC$, $AK$ be ...
anonimo's user avatar
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Hard geometry problem: given a triangle with sides 16, 30, 34, find the area of the triangle introduced by the inscribed circle

You are given a triangle $\triangle PQR$ with sides $16, 30, 34$. Let the incircle touch the sides of $\triangle PQR$ at $X,Y,$ and $Z$. Given that the ratio $[XYZ]/[PQR]$ can be written as $\frac{m}{...
allan's user avatar
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Proof for a construction of excenters

Problem Let $I$ be the incenter of $\Delta ABC$. Produce $AI$, $BI$ and $CI$ to $J_A$, $J_B$ and $J_C$ respectively such that $J_BAJ_C$, $J_CBJ_A$ and $J_ACJ_B$ are straight lines. Prove that $J_A$, $...
Yuta's user avatar
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Area of a triangle $\triangle ABC$

Let be $\triangle ABC$ a triangle with its area of $48\text{cm}^{2}$. $M$ is the mid of $[BC]$ and $P$ is the mid of $AM$. We know that $BP \cap AC = \left\{N\right\}$. The question is: how to compute ...
Iuli's user avatar
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2 votes
1 answer
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$X_{25}$ Point in a Triangle

I saw the main problem, which I have described as Theorem 5 below, on C. Kimberling's website. I made some attempts to prove the problem with geometric methods. I wrote notes. I will present them ...
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4 votes
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How can I use nine-point circle to solve this concyclic problem

I saw this problem on the Discord Math channel. H is the orthocenter of △ABC. D, E and F are the foot of the altitudes of △ABC passing through A, B and C respectively. Lines EF and BC intersect at R....
GNUSupporter 8964民主女神 地下教會's user avatar
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Is this a sufficient amount of knowledge to define a unique ellipse?

Recently, I've been trying to work out a closed formula for the Mandart inellipse of a triangle, and I made a little plaything on Desmos to streamline the process. So far I've successfully located the ...
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What is this triangle center, and is this a valid formula for it?

Take an arbitrary triangle with vertices $A$, $B$, and $C$ with side lengths opposite to the vertices $a$, $b$, and $c$. Then, assume this triangle has no mass, and hang a series of weights ...
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Why is the distance from orthocenter to vertex twice the distance from circumcenter to opposite side? [duplicate]

In the diagram above, $$2SP=AO$$ in description : line from orthocenter is 2 times of line from circumcenter. But I remember, someone in MSE said It's Euler line (I have read Wikipedia article but ...
Man's user avatar
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1 answer
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Possible triangle center associated with Apollonius circle of excircles

When I was playing around with Geogebra, I personally found a possible triangle center, but I'm not 100 % sure if my personal conjecture is true. Consider the following configuration: Let $E_A$, $E_B$...
K. Miyamoto's user avatar
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2 answers
128 views

Show that the circumscribed circle passes through the middle of the segment determined by center of the incircle and the center of an excircle.

Show that the circumscribed circle for a triangle passes through the middle of the segment determined by the center of the incircle and the center of an excircle. I found this Incenter and ...
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How to to find coordinates of the center of a triangle in a $3$-d environment if the $x, y, z$ vertexes are known?

Data provided: $x, y, z$ coordinates for 3 points in space (it's 3 stars in the solar system). I have the stars coordinates from the Galactic coordinates system and basically I want to find the $x,y,z$...
Odin Oji's user avatar
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1 answer
121 views

Prove that in an acute triangle a line drawn through $A$ and the circumcenter is perpendicular to the reflection of $BC$ on the angle bisector of $A$

Here is a picture provided: The angle shown here is what we need to prove. This is where I got: suppose $BAC$ angle is $a$; $ABC$ is $b$; and $BCA$ is $c$. $BC$ and $B'C'$ intersect at $K$. Then $AC'...
GoldBeni's user avatar
3 votes
1 answer
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Coinciding centroids of two triangles

I recently came up with a question. Consider $\triangle ABC$. How many distinct triangles $\triangle DEF$ are there such that Centroid of $\triangle DEF$ coincides with that $\triangle ABC$ Each side ...
Centelle's user avatar
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How long can we iterate circumcenters?

Given a triangle $T$, we can define a transfinite sequence $(\mathsf{CS}_\alpha(T))_{\alpha\in\mathsf{Ord}}$ of "circumsets" as follows: $\mathsf{CS}_0(T)=T$, $\mathsf{CS}_{\alpha+1}(T)=\...
Noah Schweber's user avatar
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Will point M act as a centre of circle, and if yes. Why?

I've gotten around this problem. But fail to understand why point M will act as a centre in this problem? If NM is perpendicular to M, how does it ensures that point M will be the centre? Because the ...
Lakshay Khanna's user avatar
3 votes
1 answer
113 views

Intuition for why triangles have unique incircles

It's easy to figure out why triangles have unique circumcircles; take two points on a side and look at the family of circles passing through them, only one of which (and one of which always) passes ...
harry's user avatar
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2 votes
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Ratio in which incenter divides median

Apparently, the incentre of a triangle, if it lies on a median, divides it in the ratio $$\frac{BD}{DF} = \frac{AB+BC}{AC} $$ as per this figure(where $BF$ is the median) : Proving it for isosceles ...
harry's user avatar
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2 votes
1 answer
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Let $T$ be the centroid of a triangle $ABC$, and let $P$ be the midpoint of the side $\overline{AC}$. The line containing the point $T$ parallel to th

Question : Let $T$ be the centroid of a triangle $ABC$, and let $P$ be the midpoint of the side $\overline{AC}$. The line containing the point $T$ parallel to the line $BC$ intersects the side $\...
Rajib Biswas's user avatar
4 votes
1 answer
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Shortest distance between orthocenter and origin

Consider the parabola $y^2=4x$. Let $V$ be the vertex, $F$ be the focus and $P$ be a point on the parabola such that $T$ is the foot of perpendicular of point $P$ on the directrix. Find a point $P$ ...
DatBoi's user avatar
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4 votes
1 answer
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What is the geometrical importance of the Euler Line?

What is the geometrical importance of the Euler Line (ie, the line through the centroid, orthocenter, and circumcenter (and other points) of a non-equilateral triangle)? What is meant by importance ...
Dron Bhattacharya's user avatar
0 votes
1 answer
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Using cosine rule to find distance between circumcenter and excenter

Here we have $\triangle ABC$ with circumcenter $O$ , Incenter $I$ and excenter opposite to vertex $A$ as $E$.As the title suggests, I have to find $EO$. Now I know $OI=R\sqrt{1+8\Pi_{cyc}\sin (A/2)}$ ...
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Distances between the circumcenter, orthocenter, incenter, and nine-point center of a triangle

I recently saw in a book which said: if $O,H,I,N$ are the circumcenter, the orthocenter, the incenter and the center of the nine point circle of a triangle, $R,r,\rho$ are the radii of the ...
atzlt's user avatar
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1 vote
0 answers
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Is it possible to find the vertices of a triangle given its incenter, circumcenter and centroid?

It is well-known that for an equilateral triangle the centroid, incenter and circumcenter are all the same, and that for any non-equilateral triangle these three centers are different points. So, ...
Anton's user avatar
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1 vote
1 answer
455 views

How to calculate coordinates of equilateral triangle point from coordinate of centroids and rotation?

For $ABC$ triangle with $H$ place in the centre of $[AB]$ and $[HC] = 5$ I want to calculate the $xy$ coordinate of $ABC$ from $G$ point centroids of $ABC$ triangle and for example a rotation of $60°$ ...
MrSolarius's user avatar
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0 answers
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What do we call a point in a triangle where the length from it to the other vertices are equal?

I tried a lot of research and I still can't find it. So what do we call the middle point where the length from it to the other vertices are equal? Here's a diagram. What we need to find out Does ...
1010011010's user avatar
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1 answer
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A Euclidean Geometry Construction

I have constructed the following configuration, as shown in the image below. Click for image $ABC$ is a triangle with the circumcircle and incircle drawn, and point $O$ is the incenter. Furthermore, ...
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