# 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....
1 vote
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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 ...
1 vote
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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 ... 31 views

### 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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### 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 ...
1 vote
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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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### 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 ...
1 vote
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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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$\triangle ABC$ 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 $\... 2 votes 1 answer 310 views ### How can an angle bisector be constructed using just a pair of perpendicular straightedges (no compass) Given 3 non-collinear points A, B, C, how can the bisector of angle ABC be constructed using neither a compass nor a ruler (e.g. any form of length measurement), just pair of perpendicular ... 2 votes 3 answers 685 views ### Prove$(-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 $$x^3-3x^... 0 votes 0 answers 54 views ### For a given \triangle ABC, construct a \triangle A'B'C' that has \triangle ABC as its extouch triangle For a given \triangle ABC, construct a \triangle A'B'C', so that the \triangle ABC is the extouch triangle of \triangle A'B'C'. The extouch triangle is the triangle formed by the points of ... 0 votes 1 answer 126 views ### What's the distance between the centroid of a scalene triangle and a point on its edge, at a given angle? Suppose we have a triangle where no sides or angles are equal (scalene triangle), but assuming we know all angles and lengths of the sides. How can you calculate the distance between the centroid and ... 0 votes 1 answer 46 views ### Proving concurrency of point reflections in a triangle Let P be a point inside the triangle \Delta ABC; P_a, P_b, P_c are reflection of P around BC, CA, AB respectively. What conditions are on P such that AP_a, BP_b, CP_c are concurrent? I ... 1 vote 0 answers 44 views ### Is there a name for the 1st isodynamic point of the contact triangle? For a triangle △ABC, take the contact triangle (the pedal triangle of the incenter) △A′B′C′. Does the first isodynamic point of △A′B′C′ have a name relative to △ABC? (Is it one of the Kimberling ... 0 votes 1 answer 145 views ### Incenter of a triangle formed by three lines. How can we find the incenter of a triangle (without using its vertices) that is formed by three lines y=m_1x + c_1, y=m_2x + c_2, y=m_3x + c_3? 2 votes 2 answers 184 views ### Locus of centroid of a \triangle{ABC} with A=(\cos\alpha,\sin\alpha), B=(\sin\alpha,-\cos\alpha), and C=(1,2) If A(\cos\alpha, \sin\alpha), B(\sin\alpha, -\cos\alpha) and C(1,2) are the vertices of \triangle ABC, find the locus of the triangle's centroid as \alpha varies. Let centroid be (h,k),$$... 2 votes 1 answer 76 views ### 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://... 2 votes 0 answers 115 views ### 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}... 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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### 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.
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 ...
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 ...