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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The incentre of triangle lies on original circle

Prove that if the tangent lines from A are drawn to the circle, the incentre of triangle ABC lies on the original circle
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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....
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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 ...
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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 $E_A$, $E_B$...
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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 $x,y,z$...
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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'...
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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 ...
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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)=\...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 $P$ ...
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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 ...
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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 ...
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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, ...
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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 $60°$ ...
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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 ...
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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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Circumcircle of mid points of a triangle touching the circumcircle of the triangle

$D,E,F$ are mid points of sides $BC,CA,AB$ of triangle $ABC$ and the circumcircles of DEF,ABC touch each other then find $\sum{\cos^2A}$. The problem is that I am able to imagine a scenario where this ...
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Geometry problem - prove that incentre of two triangles coincide

$AB$ and $AC$ are tangent to the circle at $B$ and $C$ respectively. Let $OA$ and $BC$ intersect at M. Let $E$ be an arbitrary point on the circle. Extend $EM$ to meet the circle at F. (1) Prove that $...
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Convergence of Mixtilinear Triangles to a Point

First, some definitions: A mixtilinear incircle of a triangle is a circle that is tangent to two sides of the triangle and internally tangent to that triangle's circumcircle. There are three ...
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Interpretation of complex trilinear coordinates

The point $X_{5374}$ in the Encyclopedia of Triangle Centres has trilinear coordinates $$\sqrt{\cot A}:\sqrt{\cot B}:\sqrt{\cot C}$$ If the reference triangle is obtuse, one (and only one) of these ...
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Properties of an apparently new triangle centre with trilinear coordinates $\frac1{\sqrt{a\cos A}}:\frac1{\sqrt{b\cos B}}:\frac1{\sqrt{c\cos C}}$

This other question asked: In an acute triangle $ABC$ let $A_1$ and $A_2$ be the intersections of the altitude from $A$ and the circle with diameter $BC$, with $A_1$ closer to $A$. $B_1$, $B_2$, $C_1$...
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A hard geometry problem involving harmonic divisions

Let acute triangle $ABC$. Let $A_1$ and $A_2$ the intersections of the circle of diameter $(BC)$ and the altitude from $A$ to $BC$ ($A_1$ is closer to $A$ than $A_2$). Similarily define points $B_1$, $...
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How to compute a particular sum of ratios in a triangle?

In triangle $ABC,$ $AD$ is median and a secant $EF$ cuts $AD$ in the ratio $p:q$ and $G$ is fixed. We wish to show that $BE/EA + CF/FA$ is constant and is equal to $2q/p.$ I tried in the following ...
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How can I find the distance between the centres of two circles?

I'm a senior year maths student and I stumbled upon a question from a maths competition from a previous year. I seem to be on the cusp of solving it but I am unable to solve for the radius (to give me ...
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0 votes
1 answer
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3 new Points lying on Jerabek Hyperbola?

R is the circumcenter, H is the orthocenter of the triangle ABC. Then points F,G,E are the points of intersection of the altitudes with the sides of the triangle ABC. U, V, W are the intersection ...
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Finding the area of an isosceles triangle with inradius $\sqrt{3}$ and angle $120^\circ$. Different approaches give different results.

I am trying to solve a question which majorly provides these details- There is an isosceles triangle with the largest angle being $120^\circ$. The radius of its incircle is $\sqrt 3$. The question ...
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Finding orthocentre of $\Delta$ formed by intercepts of plane $3x+y+3z=9$

Question: The plane $3x+y+3z=9$ intersects the co-ordinate axes at $A, B, C$. Find orthocentre $(H)$ of $\Delta ABC$. We see $\Delta ABC$ is isoceles, so let's assume $H = (h, k, h)$, as vertices $...
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A',B',C' are midpoints of triangle sides BC, CA, AB and AD is altitude. Prove $\measuredangle B'DC'=\measuredangle B'A'C'$. [closed]

$\triangle ABC$ is a triangle and A'B'C' are the midpoints of the sides $\overline{BC}, \overline{CA}$ and $\overline{AB}$ respectively. If $\overline{AD}$ is the altitude through $A$, prove that $\...
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2 votes
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How can an angle bisector be constructed using just a pair of perpendicular straightedges (no compass)

Given 3 non-collinear points A, B, C, how can the bisector of angle ABC be constructed using neither a compass nor a ruler (e.g. any form of length measurement), just pair of perpendicular ...
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Prove $(-a,-\frac{a}{2})$ is orthocenter for triangle formed by given lines

I need to prove that $(-a,-\frac{a}{2})$ is the orthocentre of the triangle formed by the lines $$y = m_ix+\frac{a}{m_i}$$ with $i = 1,2,3$; $m_1,m_2,m_3$ being the roots of the equation $$x^3-3x^...
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For a given $\triangle ABC$, construct a $\triangle A'B'C'$ that has $\triangle ABC$ as its extouch triangle

For a given $\triangle ABC$, construct a $\triangle A'B'C'$, so that the $\triangle ABC$ is the extouch triangle of $\triangle A'B'C'$. The extouch triangle is the triangle formed by the points of ...
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What's the distance between the centroid of a scalene triangle and a point on its edge, at a given angle?

Suppose we have a triangle where no sides or angles are equal (scalene triangle), but assuming we know all angles and lengths of the sides. How can you calculate the distance between the centroid and ...
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Proving concurrency of point reflections in a triangle

Let $P$ be a point inside the triangle $\Delta ABC$; $P_a, P_b, P_c$ are reflection of $P$ around $BC, CA, AB$ respectively. What conditions are on $P$ such that $AP_a, BP_b, CP_c$ are concurrent? I ...
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Is there a name for the 1st isodynamic point of the contact triangle?

For a triangle △ABC, take the contact triangle (the pedal triangle of the incenter) △A′B′C′. Does the first isodynamic point of △A′B′C′ have a name relative to △ABC? (Is it one of the Kimberling ...
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Incenter of a triangle formed by three lines.

How can we find the incenter of a triangle (without using its vertices) that is formed by three lines $y=m_1x + c_1, y=m_2x + c_2, y=m_3x + c_3$?
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2 votes
2 answers
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Locus of centroid of a $\triangle{ABC}$ with $A=(\cos\alpha,\sin\alpha)$, $B=(\sin\alpha,-\cos\alpha)$, and $C=(1,2)$

If $A(\cos\alpha, \sin\alpha)$, $B(\sin\alpha, -\cos\alpha)$ and $C(1,2)$ are the vertices of $\triangle ABC$, find the locus of the triangle's centroid as $\alpha$ varies. Let centroid be $(h,k)$, $$...
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Is this point regarded as a triangle center from ETC standard?

Given a reference triangle ABC, create the cevian triangle A’B‘C’ of the symmedian point K of ABC. Then what is the point K called when the reference triangle is A’B‘C’? I search the ETC (https://...
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Difficult Trigonometry Concurrency

Problem Statement In arbitrary $\Delta ABC$, $I$ is the incenter. $D,E,F$ are feet of perpendicular from $I$ on $\overline{BC}, \overline{CA}, \overline{AB}$. $D',E',F'$ are on $\overrightarrow{ID}...
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1 vote
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Coordinates of centres of central triangles with respect to the reference triangle

In Kimberling's Encyclopedia of Triangle Centers, a lot of centres are described as the centres of certain central triangles of the reference triangle, whether as a main or alternate definition. For ...
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9 votes
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Eliminating unwanted branches of algebraic curves related to triangle centres

Lately I have become fascinated with triangle centres. To that end, I have written a small Python module that can compute explicit positions of centres for arbitrary triangles in the plane to ...
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GRE geometry questions about finding the angle between a side of a triangle and a circumradius

I am struggling with reconciling the fact that all the middle lines are the same length with the fact that the angles aren't the same.
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3 votes
1 answer
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Significance of Equal Angle Triangle Center

While playing around with triangle centers and came across one I did not know, the center where each triangle corner heading is equally spaced ($120^\circ$ spacing). Does this specific triangle ...
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17 votes
1 answer
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An interesting geometry problem about incenter and ellipses.

Let $I$ be the incenter of a triangle $ABC$. A point $X$ satisfies the conditions $XA+XB=IA+IB$, $XA+XC=IA+IC$. The points $Y,Z$ are defined similarly. Show that the lines $AX,BY,CZ$ are concurrent or ...
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