# 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.

50 questions
Filter by
Sorted by
Tagged with
17 views

### 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
• 1
69 views

### 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
62 views

### 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 ...
• 1,137
42 views

### 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 ...
• 1,137
97 views

### 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 ...
• 89
1 vote
46 views

### 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 $E_A$, $E_B$...
55 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 ...
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 $x,y,z$...
55 views

• 220k
38 views

### 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 ...
51 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 ...
• 988
1 vote
81 views

### 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 ...
• 988
298 views

• 1,173
190 views

### 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 ...
• 1,613
1 vote
70 views

### 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 ...
• 89.9k
89 views

### 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 $A$. $B_1$, $B_2$, $C_1$...
• 89.9k
127 views

• 306
213 views

1 vote
104 views

### 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 ...
• 89.9k
115 views

### 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 ...
• 89.9k
86 views

### 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.
• 345
115 views

### 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 ...
• 31
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 ...