For questions about properties and applications of triangles

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0
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1answer
13 views

Pendulum tension force

I realize this is physics related, although the problem is really about math so I thought it would be a good fit for this site. My illustration is supposed to depict a pendulum and the forces ...
2
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1answer
54 views

Combinatorics - Integer sided triangles with integer median

The original problem states: "Given a number N, how many integer-sided triangles $(a,b,c)$ with an integer median $m_{c}$ exist, provided that $a \leq b \leq c \leq N$?". I've managed to get it down ...
0
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1answer
9 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
2
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0answers
28 views

Interpretation of median length for an invalid triangle

Background: My very first and naive take on the Project Euler problem 513 went wrong, as I counted also triples violating the triangle inequality. Many formulas return an invalid result for an ...
1
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1answer
30 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
3
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0answers
30 views

Tripartite n+1-regular graph containing a triangle

Suppose a tripartite, $(n+1)$-regular graph. Each one of its $3$ parts $(A,B,C)$ contains $n$ nodes. Show that the graph contains a triangle. I think the fact that it is $n+1$ and not $n$ plays an ...
3
votes
1answer
338 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
0
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1answer
36 views

Solving triangle

If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of ...
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2answers
52 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
0
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1answer
24 views

Given sides and a bisection, find angles in a triangle

Consider a triangle $ABC$ where the angle $A$ is $60^{\circ}$. Draw its bisection intersecting $BC$ at $D$. Let $AB = x$, $BD = y$ and $AC=x+y$, $\angle ABC = \alpha$ and $\angle ACB = \beta$. Find ...
2
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1answer
33 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
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2answers
22 views

Trigonometry confusion with triangle in weird question

I was wondering how do you get x from the triangle below:
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0answers
8 views

Determine the angles of the obtuse angles triangle formed by the vertices [on hold]

Determine the angles of the obtuse angles triangle formed by the vertices (1,6),(8,1),(9,-4) Acute angles between two lines
0
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0answers
20 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
1
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1answer
311 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Any ideas on how to do this?
8
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5answers
2k views

A Math Olympiad question regarding Geometry

A little bit of a backstory (you may skip this if you want): My high school math teacher knows that I love math, but he also knows that I usually drift off during my classes, perhaps because it's too ...
1
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5answers
59 views

Find third point to make isosceles triangle with a specific area

Using points A(1,2) and B(-2,-2), find a third point, with a positive y-value, that makes ABC an isosceles triangle with area 10 units${^2}$. I have found AB to be 5 and used this as $r^2$ below.. ...
2
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4answers
45 views

Find the type of triangle from equation.

In triangle $ABC$, the angle($BAC$) is a root of the equation $$\sqrt{3}\cos x + \sin x = \frac{1}{2}.$$ Then the triangle $ABC$ is a) obtuse angled b) right angled c) acute angled but not ...
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0answers
14 views

How to find local extrema of f (p) give us area of triangle A1B1C1

For a right triangle ABC ( angle C = 90) on the rights height CC1 is chosen point P and consider the triangle A1B1C1 (A1 = AP cross BC, B1 = BP cross AC), if p is distance from point P to AB, to find ...
25
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4answers
1k views

Two circles inside a right angled triangle!

The other day I was playing with Ms Paint drawing circles here and there - I coincidentally drew a circle inside a right angled triangle which I already drew. Strangely A problem struck to my mind ...
1
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1answer
35 views

Pascal's triangle

I was out sick for a while (2 weeks) and just got back and now we are doing whatever this is! Can someone explain to me what this is or show me a video on how to do it? "Use Pascal's triangle and the ...
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3answers
2k views

How do I map a 3D triangle into 2D?

The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in x,y,z form, and want to map them onto the plane described by the normal of the triangle, such that I end up ...
0
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1answer
25 views

finding the value of a node in Pascal’s (a.k.a Yanghui's) triangle [closed]

Image the Pascal Triangle is on an x-y cartesian plane. so that the values of the nodes, by location are ...
8
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1answer
146 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
0
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1answer
22 views

Find length of side of a triangle.

Let $ABC$ be a right angled triangle with $BC = 3, AC = 4$. Let $D$ be a point in the hypotenuse $AB$ such that $\angle{BCD} = 30^\circ$. Find the length of $CD$. I found $AB = 5$. How do we find ...
3
votes
1answer
67 views

Locus of a point $P$ inside $\triangle ABC$

$P$ is a point inside $\triangle ABC$. $X$, $Y$ and $Z$ are feet of perpendiculars from $P$ on $BC$, $CA$ and $AB$ respectively. Find the locus of $P$ if $XY=XZ$ and $A \equiv (4,3)$, $B \equiv ...
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vote
3answers
57 views

Find circle radius by given triangle inside

So the triangle inside the circle: $AB = 9$cm $CB = 6$cm $CH = 5$cm I think solving this problem involves similar triangles. Thanks in advance, I'd like to have a solution suitable for 9th ...
15
votes
4answers
200 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
0
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0answers
20 views

How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$ for an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively

Let $ABC$ is an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively. Let $CM\cup NP=X, AN\cup MP=Y, BP\cup NM=Z$. How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$? ...
9
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3answers
512 views

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. Why?

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please ...
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1answer
331 views

If an equilateral triangle has an area of 36 units squared, what is the length of a side to the nearest tenth?

I have been working with finding the area of a regular triangles, squares, and hexagons using special right triangle formulas drawn from the radii and apothems, but I cannot for the life of me work ...
0
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1answer
910 views

Find coordinates of vertex in right triangle

I have a right triangle with known points $A(x_1,y_1), B(x_2,y_2)$ and known cathetus $AC$ and $BA$. I need to find the coordinates of point $C$.
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3answers
313 views

Geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
2
votes
1answer
55 views

Ratio of sides of triangle $ABC$

If in a triangle $\Delta ABC$ with $a$, $b$ and $c$ as sides $$\begin{align}\left(Cot\frac{A}{2}\right)^2 ...
0
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1answer
18 views

Bounding inradius, given circumradius.

The problem in my book is as follow. In a $\Delta ABC$ , if $r=r_2+r_3-r_1$ and $\angle A >\dfrac{\pi}{3}$ , then the range of $\dfrac{s}{a}$ is equal to: (Here $r_i $ are exradii) I used ...
0
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2answers
197 views

How long is side $a$ in this right triangle if $a = b$, given $c$?

A right angled triangle has side lengths labeled as so. A common geometric construction that shows three squares sitting upon the sides of a right triangle with lengths A, B, and C However unlike in ...
1
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2answers
47 views

Find distance between two poles.

2 poles, AB of length 2 metres and CD of length 20 metres are erected vertically with bases at B and D. The two poles are at a distance not less than twenty metres. It is observed that tan(angle(ACB)) ...
1
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1answer
22 views

Finding coordinates of the third point of a triangle from given?

In ABC triangle we know the coordinates of A and B vertices. We also know lengths of 2 edges shown in the picture and the third edge is calculatable. What is the most efficient functon to find x3 and ...
0
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0answers
17 views

What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
2
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2answers
107 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
5
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2answers
125 views

Three circles having centres on the three sides of a triangle

NOTE: I would appreciate it if you provided a hint and not the whole solution. BdMO 2014 Nationals: In acute angled triangle ABC, considering a portion of side BC as diameter a circle is drawn ...
2
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2answers
119 views

Beautiful triangle problem

Circle, inscribed in $ABC$, touches $BC, CA, AB$ in points $A', B', C'$. $AA' BB', CC'$ intersect at $G$. Circumcircle of $GA'B'$ crosses the second time lines $AC$ and $BC$ at $C_A$ and $C_B$. Points ...
2
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2answers
259 views

Centroids of triangle

On the outside of triangle ABC construct equilateral triangles $ABC_1,BCA_1, CAB_1$, and inside of ABC, construct equilateral triangles $ABC_2,BCA_2, CAB_2$. Let $G_1,G_2,G_3$ $G_3,G_4,G_6$be ...
3
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2answers
163 views

Triangle geometry - synthetic proof

I'm looking for a nice synthetic proof of the following fact. Consider a non-isosceles triangle, pick a vertex. Assume that the median and the altitude passing through this vertex are isogonal ...
11
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2answers
3k views

Equilateral triangle inscribed in a triangle

Consider triangle ABC. Point D is on AC, E is on BC and F is on AB. Given that triangle DEF is equilateral and that segments CD, BE and AF are equal in length, prove that triangle ABC must also be ...
2
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1answer
304 views

Triangle problem related to finding an area

Given a triangle $\triangle ABC$ . Points $P, Q, R$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively. $\overline{AP}$ bisects $\overline{BQ}$ at point $X$, ...
4
votes
1answer
88 views

Nine-point circle equivalent for tetrahedrons?

Nine-point circle for a triangle is defined as the circle that passes through: the midpoint of each side the foot of each altitude the midpoint of the line segment from each vertex to the ...
2
votes
3answers
94 views

Proving $ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}$ in a geometric context

Prove or disprove $$ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}. $$ I have no idea where to start, but it must be a simple proof. Trivia. This fact was used for determination of resistance of two ...
0
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1answer
371 views

Calculate 3rd point of a triangle, given 2 points and all angles in 2D

I have stumbled upon an interesting problem. I tried to find an answer here but there are just too many similar threads which did not really help me, so I was trying to figure it out by myself. The ...
0
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1answer
10 views

How to get a Right Triangle's points' coordination in the space?

I have a Right Triangle with equal legs of 1 unit long rotated on 3 individual angles in the space like in the picture below: As could be seen in the picture, the input I have are the angles 'a' ...