For questions about properties and applications of triangles

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3
votes
2answers
800 views

how many rectangles in this shape

I've learned in my high school the solution to such riddle: How many rectangles are there in this shape: the solution is through combinations: in this shape is a $5\times 6$ grid so the number of ...
3
votes
3answers
162 views

Find out the angle of <ABC

Help me to solve it please.how could it be done.I tried but nothing comes out.Help me please
2
votes
3answers
5k views

Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.

I am writing a program and I need to calculate the 3rd point of a triangle if the other two points, all sides and angles are known. ...
8
votes
2answers
127 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
5
votes
1answer
135 views

If this relation holds, then is the triangle equilateral?

Let $ABC$ be a triangle. If $$\sum_{cyc}\frac{BC}{4AC\cos^2({\frac{\angle BAC}{2})}+BC}=\frac{3}{4}$$ then the triangle is equilateral? We can check if we set $\widehat{BAC}=\pi/3$ and $AB=BC=CA$ that ...
5
votes
3answers
836 views

Largest Triangle with Vertices in the Unit Cube

How would one find a triangle, with vertices in or on the unit cube, such that the length of the smallest side is maximized? And what is that length? A lower bound for the length is $\sqrt{2}$, by ...
4
votes
2answers
864 views

Find an angle of an isosceles triangle

$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use ...
4
votes
4answers
943 views

Constructing a triangle given three concurrent cevians?

Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, ...
4
votes
3answers
4k views

Find the coordinates in an isosceles triangle

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
3
votes
2answers
103 views

Proof of a geometric statement

If $D$ is a point inside a triangle $\triangle ABC$ then how the following statement is true. statement: $AB+AC>BD+DC$. I have tried in the following way but it seems to me defective. ...
2
votes
1answer
78 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 ...
2
votes
3answers
791 views

Proving the length of angle bisector

How do I prove that a triangle with sides a, b, c, has an angle bisector (bisecting angle A) is of length: $$\frac{2 \sqrt{bcs(s-a)}}{b+c}$$ I have tried using the sine and cosine rule but have ...
2
votes
1answer
331 views

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva's theorem. Is there a ...
2
votes
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 $X$, ...
1
vote
1answer
2k views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
0
votes
3answers
54 views

Point inside the area of two overlapped triangles

The question is as simple as that, but I have been trying to figure out an answer (and searching for it) with 0 results. I mean, given two triangles (in 2D) I want to find just a single point which ...
0
votes
3answers
119 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
0
votes
1answer
132 views

Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Let a, b, c be the three side lengths of a triangle. Prove that $$\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\geq 3$$ Under what conditions is equality obtained?
0
votes
1answer
2k views

maths - find vertices when 1 vertex and center point is given in polygon

I want to know if there is any general formula to find out vertices (co-ordinates) of a polygon (3 or more equal sides) when following is given: ...
8
votes
1answer
247 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
7
votes
2answers
73 views

Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...
7
votes
2answers
702 views

Problem with the Pythagorean theorem [duplicate]

The Pythagorean theorem has already been proved and it is a basic fact of math. It always works, and there are proofs of it. But I have found a problem. Say you want to get from point ...
5
votes
4answers
144 views

Is there an integer that $\sqrt{3}$ can be multiplied by that will produce a whole integer?

The question came up while messing around with graph paper. I wanted to make an isosceles triangle where the length of one side and it's hight were both integers. The closest I could get was a base ...
4
votes
2answers
64 views

How do I find the Intersection of two 3D triangles?

I've got a rather complicated geometry problem that I'm trying to solve - how to find the intersection between two triangles in 3D space. I've looked around at other questions and answers on this site ...
4
votes
2answers
250 views

Does “triangle” in English exclude degenerate triangles?

Just for fun read few problems on the projeteuler.net website. Number 276 found interesting: Consider the triangles with integer sides a, b and c with a ≤ b ≤ c. An integer sided triangle ...
4
votes
2answers
455 views

Proving $\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1$

I was stumped by another past-year question: In $\triangle ABC$, prove that $$\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1.$$ Here's what I have done so far: I tried to replace $C$, using ...
4
votes
1answer
1k views

Finding the distance between two gears

I have the following problem: In my class, we did a majorly complicated method to figure this out but I think there is a better way to do this... Here is the exact problem: A belt fits snugly ...
3
votes
2answers
66 views

In triangle $ABC$, $a^2+c^2=3b^2$

In triangle $ABC$, we have $a=BC$, $b=CA$ and $c=AB$ as usual. What is a necessary and sufficient condition for $a^2+c^2=3b^2$ to hold? I created this problem as a generalization of $a^2+c^2=2b^2$ ...
3
votes
3answers
143 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
3
votes
2answers
3k views

How do you find the height of a triangle given $3$ angles and the base side? Image given.

This question has me absolutely stumped. This is the image of the question, how can I work out $x$? I've been doing a variety of attempts but I just cant get it.
3
votes
2answers
9k views

Finding the area of triangle if length of medians are given

My question is: In triangle ABC ,length of median from vertex A is $13$ , length of median from vertex B is $14$ , length of median from vertex C is $15$. Compute the area of triangle ABC.
2
votes
1answer
55 views

Given a particular triangle that has been constructed, I want to prove that one of the angles must be $> 45$ degrees. [duplicate]

Suppose you are given an acute triangle $XYZ$ with the following properties: At $\angle XZY$, the $\angle$ bisector is drawn and extended all the way to $XY$. Lets call the point where it intersects ...
2
votes
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
votes
2answers
184 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
2
votes
1answer
131 views

Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. ...
2
votes
3answers
1k views

How many triangles can be created from a grid of certain dimensions?

How would you determine how many non-degenerate triangles can be drawn by connecting points in a $5 \times 5$ grid?
2
votes
1answer
471 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
2
votes
1answer
72 views

find parameter for maximize area

suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are $A(cos(3-t),sin(3-t))$ $B(cos(t),sin(t))$ ...
2
votes
1answer
564 views

Move Point A along a line

Sorry, can't post images if my rep is below 10, and can't post more than 2 links. I removed the http section so it won't count as a link. I hope this isn't against forum rules, I'm not hurting anyone. ...
2
votes
1answer
3k views

What's the ratio of triangles made by diagonals of a trapezoid/trapezium?

In the above image, what will be the ratio of areas of triangle $A$ and $B$? From Googling, I've found that: $\operatorname{Ar}(A) = \dfrac{a^2h}{2(a+b)}$ and $\operatorname{Ar}(B) = ...
2
votes
3answers
470 views

Geometry - Equilateral triangle covered with five circles

I have to cover an equilateral triangle (whose sides are 1m long) with 5 identical circles: what's the minimum radius of the circles?
2
votes
1answer
32k views

How do I find the angles of a triangle if I only have the lengths of the sides?

Is it possible to find the angles of a triangle if I only have its sides? If so, how can I achieve this? Regarding my knowledge of triangles: Whilst I was taught trigonometry a few years ago, I ...
1
vote
1answer
53 views

Proof of a set of triangles and unit squares

Suppose that there is $S$, a finite set of unit squares. So, $S$ is chosen from a larger grid of unit squares. The unit squares of $S$ are tiled with isoceles right triangles. Each of these triangles ...
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vote
4answers
56 views

Proving $\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$

As the title, By considering $\bigtriangleup$ABC, Prove $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$$ Thanks
1
vote
1answer
81 views

How to calculate number of triangles and points after dividing a triangle n times?

When having a triangle and dividing it n times, how to get the number of triangles and points? ...
1
vote
1answer
24 views

What are the limitations of non-metric distances?

If the triangle inequality does not hold for a distance function (i.e. it is not metric), will this limit its application in some area?
1
vote
1answer
43 views

What is the range of angle in front of longest triangle edge?

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?
1
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4answers
195 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
1
vote
3answers
220 views

Prove that point M is on circle c

It's hard to create question names that make sense. Anyhow, the following is another question from my math assignment. Line-segment AB has a fixed length of 10 units. point A moves on the positive ...
1
vote
2answers
989 views

What is the number of triangles with integer sides, given the length of the longest side?

Suppose $a,b,c \in\mathbb N$, and the value of $c$ is known and fixed, while $a$ and $b$ are unknown and are both smaller than $c$. What is the total number of unique triangles possible with $a, b$ ...