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Questions tagged [triangles]

For questions about properties and applications of triangles.

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2answers
53 views

Prove $a^2b(a-b)+b^2c(b-c)+c^2a(c-a) \geq 0$

Given the sides of a triangle a, b, c, prove that $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a) \geq 0.$$ I can't seem to figure out how to simplify it. I assumed that $a$ is greatest out of the three, but I can't ...
2
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4answers
65 views

Proving that, for an acute $\triangle ABC$, $\sin A + \sin B+\sin C\gt \cos A+\cos B+\cos C$

I need to prove or disprove that in any acute $\triangle ABC$, the following property holds: $$\sin A + \sin B + \sin C \gt \cos A + \cos B + \cos C$$ To begin, I proved a lemma: Lemma. An ...
5
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3answers
62 views

When is the Euler line parallel with a triangle's side?

When is the Euler line parallel with a triangle's side? I have found that a triangle with angles $45^\circ$ and $\arctan2$ is a case. Is there any other case? >
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3answers
68 views

I Need Help in a Challenge [on hold]

My teacher challenged me with the question below: $$\sqrt{\frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2}\ast \left ( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \frac{\sqrt{41}}{1} \right )\ast \left ( \frac{\...
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1answer
53 views

Question of Triangle of Geometry

In triangle ABC,we have $AB>AC$ . If A' is the mid point of BC,AD is the altitude through A and if the internal and external bisectors of angle A meet at BC at X and X' respectively ,prove that $A'...
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1answer
23 views

Number of isosceles triangles formed with 12 equally spaced points lying on the circumference of a circle [on hold]

How many isosceles triangles can be formed with 12 equally spaced points lying on the circumference of a circle? The answer should be 52, but I have no idea how to solve it. Please help!
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2answers
24 views

How large is the distance between centroids of two equilateral triangles [on hold]

I have a problem. I have a triangle grid with a lot of equilateral triangles. Now I want to know what the distance is between two centroids. How can I do that? Here is a sample image of the grid:...
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2answers
31 views

Side length of largest equilateral triangle to fit in a rectangle

I was trying to print out the largest possible equilateral triangle on a standard sheet of paper (8.5 by 11 inches) and got sidetracked into the following question: what is the maximum possible side ...
0
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0answers
29 views

How to calculate area of triangle-like structure of blocks? (Pick's Theorem seems insufficient.)

Given the following structure, is there a formula to calculate the number of blocks? (EDIT: and I am really looking for a solution for any BASE and HEIGHT.) At first, it would seem that this is a ...
1
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0answers
24 views

Conic in Trilinear Coordinates

I have the following equation of a conic in trilinear coordinates: $$x^2+y^2+z^2-\frac{\alpha^2+\beta^2}{\alpha\beta}xy-\frac{\beta^2+\gamma^2}{\beta\gamma}yz-\frac{\gamma^2+\alpha^2}{\gamma\alpha}zx=...
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1answer
32 views

Help with proving the following points are collinear

Let BC be the shortest side of $\triangle$ABC. Let P be a point in AB such that $\angle$PCB=$\angle$BAC and Q be a point in AC such that $\angle$QBC=$\angle$BAC. Prove that the line that passes ...
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1answer
34 views

Plane Geometry Triangles [closed]

Prove that If X is any point on BC of triangle ABC , then either AB or AC greater than AX . (Reference- Pre College mathematics)
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1answer
31 views

Triangles Chapter of Plane Geometry [closed]

If AD is the altitude through A of triangle ABC, prove that AB > AC, AB = AC or AB < AC according as BD > DC.,BD = DC or BD < DC
2
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0answers
33 views

Given a triangle ABC and a square, explain why there exists 3 points P,Q,R on a square such that triangle ABC is similar to triangle PQR [closed]

Given a triangle ABC and a square, explain why there exists 3 points P,Q,R on a square such that triangle ABC is similar to triangle PQR How do I start this? I tried drawing the pictures but it just ...
0
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0answers
34 views

Get point in dot product of multiple points [closed]

I'm trying to get a dot product of a point between multiple points in a graph, i get one line in a dot product but I didn't manage to get it in multiple points, is this even possible?! The idea was to ...
3
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1answer
46 views

Show that a triangle is equilateral

A circle crosses the sides of a triangle, dividing each of them into three equal parts. Prove that the triangle is equilateral. I think that the best way is to show that $\angle BAC = \angle ABC$, ...
2
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1answer
63 views

Equilateral triangles on the sides of a triangle

We have a triangle. We then construct three points outside of the triangle by drawing three equilateral triangles on the sides of the original triangle. Now we want to do the opposite: from the three ...
9
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1answer
58 views

Semiperimeter of isosceles Heronian triangles.

A Heronian triangle is a triangle with integer sides and area, named after Heron's formula which states that the area of a triangle with sides $a$, $b$, and $c$ is $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$ ...
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0answers
25 views

Compute the radius and the central coordinate (x, y) of a circle constructed by three given points on the plane surface

I need you to explain the mathematics behind the code bellow. What is s, what are those formulas for px and py and generally, what logic are we following to find the answer here? ...
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2answers
25 views

What is measurement of angle D? [closed]

Angle 1= Angle 2, Angle 3 = Angle 4, Angle A = 90º What is the measurement of Angle D?
0
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3answers
53 views

System the system with $3\sin a+4 \cos b = 6$ and $4 \sin b+3 \cos a=1$

You have these two equations (a, b, c are angles of a triangle): $$ 3\sin(a)+4\cos(b)=6\\ 4\sin(b)+3\cos(a)=1 $$ The triangle part limits a, b, c to (0, $\pi$) and provides an additional equation: $$...
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2answers
27 views

GRE Geometry — Overlapping Circles with an enclosed figure.

Hi I'm having trouble solving this problem from a quantitative reasoning question on the GRE. The diagram and the word problem are shown above. I think I'm lost as to the properties a parallelogram or ...
0
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1answer
17 views

Given vectors AB, BC, and BC Simplify the following expression.

Given vectors $\overrightarrow{AB}, \overrightarrow{DC},$ and $ \overrightarrow{BC}$, simplify $\overrightarrow{AB} − \overrightarrow{DC} + \overrightarrow{BC} $ Here's what I have done: $\...
0
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1answer
14 views

Nine point Circle right angled and tangent

I have been working on the following tasks for some time now and I do not know how to solve it. $(A, B, C)$ is a triangle and $K$ is his nine point circle. Show it: a) $(A, B, C)$ is right-angled if ...
1
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3answers
39 views

ABC triangle with $\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$

I have a triangle ABC and I know that $\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$, where $a,b,c$ are the sides opposite of the angles $A,B,C$. Then this triangle is: a. Equilateral b. Right ...
0
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0answers
55 views

Triangle geometry with bisectors

In triangle $ABC$, $AB=10$, $CA=12$. The bisector of ∠B intersects $CA$ at $E$, and the bisector of ∠C intersects $AB$ at $D$. $AM$ and $AN$ are the perpendiculars to $CD$ and $BE$ respectively. If $...
3
votes
2answers
97 views

In a triangle, prove that $ \sin A + \sin B + \sin C \leq 3 \sin \left(\frac{A+B+C}{3}\right) $

Prove that for any $\Delta ABC$ we have the following inequality: $$ \sin A + \sin B + \sin C \le 3 \sin \left(\frac{A+B+C}{3}\right) $$ Could you use AM-GM to prove that?
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0answers
34 views

Is there a formula to calculate the length of line segment $x$ where $x$ branches off from an angle of a triangle?

It's best shown by an image: (image) where the angle $A$, $B$, and $C$ can be any known angle and the lengths of the line segments that make up the triangle can be any known length and line $x$ ...
0
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1answer
24 views

Show that MN and MP are angle bisectors

A $\triangle ABC$ is drawn ($\angle C = 90^\circ$), in which $CL$ $(L \in AB)$ is bisector. The circle $k$ with diameter $CL$ intersects AB, BC and CA, respectively, in $M$, $N$ and $P$. Show that $...
6
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7answers
247 views

Different methods give different answers. Let A,B,C be three angles such that $ A=\frac{\pi}{4} $ and $ \tan B \tan C=p $.

Let $A,B,C$ be three angles such that $ A=\frac{\pi}{4} $ and $ \tan B \tan C=p $. Find all possible values of $p$ such that $A,B$ and $C$ are angles of a triangle. case 1- discriminant We can ...
0
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1answer
19 views

Better way to find the angle subtended by OI at the vertex.

Better way to find the angle subtended by OI at the vertex A. O is circumcentere and I is incenter of a triangle ABC. So I did solved this problem by finding the distance OI which is $ \sqrt{ R^2 -...
1
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1answer
27 views

The inradius in a right angled triangle with integer side is r, if r = 4 the greatest perimeter is ??

The inradius in a right angled triangle with integer side is $r$, if $r = 4$ the greatest perimeter is ?? My attempt- I know that $ r = (s-a)\tan \frac{A}{2} $ Thus $ 2r = a+b-c $ I also know ...
0
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2answers
37 views

Distance between circumcentre and incenter of an isosceles triangle with base angle less than 45°.

Let $ABC$ be an isosceles triangle with inradius $r$, circumradius $R$ and base angle $\alpha$. The question is to find the distance between circumcentre and incenter. I know that the distance ...
0
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0answers
27 views

Need help on a problem on trigonometry

In triangle ABC, AB=10, CA=12. The bisector of ∠𝐁 intersects CA at E, and the bisector of ∠𝐂 intersects AB at D. AM and AN are the perpendiculars to CD and BE respectively. If MN=4, then find BC. ...
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1answer
45 views

Find The X In This Shape. [closed]

This is a rather weird shape that even my teacher couldn't solve. (Lets ignore the faulty touchscreen) The red letters are the angles and A.C.E are perfectly straight (Colinear) Is there an way to ...
0
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3answers
35 views

How many triangles are there with whole number side?

If $a=29$, and $b=21$, how many triangles are there such that side $c$ is a whole number? My approach: Tried using certain equations to establish relationship between sides to maybe point to right ...
0
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1answer
36 views

Need help on a trigonometry problem

The question is- Points D and E divide equal sides AC and AB of an equilateral triangle ABC according to the ratio of 𝑨𝑫: 𝑫𝑪 = 𝑩𝑬: 𝑬𝑨 = 𝟏: 𝟐. Edges BD and CE meet at point O. Find ∠𝐀𝐎𝐂. ...
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0answers
15 views

Finding the ratio between two lines on a triangle

In the obtuse triangle ABC with ∠C > 90◦ , E and F are points on the side AB such that AE = EF = F B. D is a point on the line BC such that BC is perpendicular to ED, AD is perpendicular to CF and ∠CF ...
0
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0answers
18 views

Let A be a point inside a regular polygon of 10 sides. Let $P_1, P_2,\ldots, P_{10} $ be the distances of A from the sides of the polygon.

Let $A$ be a point inside a regular polygon of 10 sides. Let $P_1, P_2,\ldots, P_{10} $ be the distances of $A$ from the sides of the polygon. If each side is of length $2$ units, then find the value ...
0
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1answer
41 views

Geometry problem based on triangles

Consider a right angled triangle $ABC$ , with right angle at $C$,$ <CAB=\theta$ and $|AC|=1$. $D$ is a point on $AB$ such that $|AD|=|AC|=1$, and $E$ is a point on $CB$ such that $<CDE=\theta$, ...
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1answer
58 views

Triangle Dilation by Radius Distance

I'm attempting to follow the accepted solution here: However, I cannot seem to get $k$ to equal out to the correct scalar value. In my use case I made up a triangle of the following 3 points ${(0,0),...
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2answers
49 views

Geometry problem (Inscribed angle theorem, circumscribed circle)

Let A and B be two different points. Show that the points P are such that the angle APB is 90 degrees and creates a circle. Decide the the radius and mid point of the circle. I have problems proving ...
0
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0answers
24 views

Radius of a circle in terms height of right triangle

Here are the constants and the arc length s=R*theta is considered constant Given only H and the arc length between theta How can you find the radius of a circle? I was able to find a relationship ...
0
votes
0answers
11 views

Triangle Intesection Test with Shared Edge

Given two triangles $T_0 = (v_0, v_1, v_2), T_1 = (v_0, v_1, v_3)$, I would like to test if there exists an overlap (area of intersection $>0$). Note, that the two triangles share a common edge $(...
0
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2answers
66 views

What is wrong with my solution of maximum value of $ \sin \frac {A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2} $ in a triangle ABC?

What is wrong with my solution of the maximum value of $\displaystyle\sin \frac {A}{2} + \sin \frac{B}{2} + \sin \frac{C}{2}$ in a triangle ABC? I am NOT after the answer. I know that $\...
1
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1answer
44 views

The side of a triangle inscribed in a given circle subtends angles $a, b, $ and $ y$ at the center.

The side of a triangle inscribed in a given circle subtends angles $a, b,$ and $y$ at the center. The minimum value of the arithmetic mean mean of $ \cos (a+ \frac{\pi}{2}), \cos(b+\frac{\pi}{2})$ ...
3
votes
1answer
37 views

Triangle centers Orthocenter

An acute $\triangle ABC$, inscribed in a circle $k$ with radii $R$, is given. Point $H$ is the orthocenter of $\triangle ABC$ and $AH = R$. Find $\angle BAC$. (Answer: $60^\circ$) $AD$ $-$ ...
0
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1answer
22 views

Interesting Property of Tri-Rectangular Tetrahedron [duplicate]

For starters, we know that a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. Here's an interesting property I stumbled upon, of which I look ...
1
vote
3answers
82 views

What is the area of a triangle with sides $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$?

I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$. After some algebra hell trying to work with Heron's formula, I ...
0
votes
1answer
47 views

Find triangle with given orthocenter and an apex on a circle

We are given $k(O; r)$, $A \in k$ and $H : OH>r$. Find points $B$ and $C$ ($B, C\in k$) such that $H$ will be the orthocenter of $\triangle ABC$. I am trying to see what is the point $F$ ($AH \cap ...