# Questions tagged [triangles]

For questions about properties and applications of triangles.

4,091 questions
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### 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 ...
1answer
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### 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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1answer
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### 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 ...
1answer
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### 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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### 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
0answers
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 $\angle BAC = \angle ABC$, ...
1answer
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### 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 ...
1answer
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### 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 $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ ...
0answers
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### 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? ...
2answers
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### 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?
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### 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 $x$ ...
1answer
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### 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 ...
2answers
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### 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 ...
0answers
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### 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. ...
1answer
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### 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 ...
3answers
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### 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 ...
1answer
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### 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. ...
0answers
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### 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 ...
0answers
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### 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 ...
1answer
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### 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 $<CDE=\theta$, ...
1answer
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2answers
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