For questions about triangles
43
votes
10answers
5k views
What's a proof that the angles of a triangle add up to 180°?
Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point.
However, now that I'm in university, I'm not ...
23
votes
6answers
7k views
How many triangles are there?
The question is how many triangles are there in the following picture?
I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I ...
15
votes
4answers
585 views
probablity of random pick up three points inside a regular triangle which form a triangle and contain the center
what is the probablity of random pick up three points inside a regular triangle
which form a triangle and contain the center of the regualr triangle
the three points are randomly picked within the ...
12
votes
3answers
839 views
Proving Stewart's theorem without trig
Stewart's theorem states that in the triangle shown below,
$$ b^2 m + c^2 n = a (d^2 + mn). $$
Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
10
votes
3answers
138 views
For which n are there primitive Pythagorean triples with legs of lengths a and a+n?
For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there ...
9
votes
1answer
785 views
The Ellipse Problem - finding an ellipse inside a triangle
The problem statement is as follows:
A triangle is dissected into six smaller triangles by its angle bisectors. Prove that the intersections of the angle bisectors of each of these smaller triangles ...
9
votes
3answers
812 views
how to prove DEF is an equilateral triangle?
ABC is an equilateral triangle，and AD = BE = CF，Prove DEF is an equilateral triangle.
8
votes
1answer
304 views
Is there a value for $\pi$ that relates to triangles?
So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which ...
8
votes
2answers
445 views
Sangaku: Show line segment is perpendicular to diameter of container circle
"From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that ...
7
votes
3answers
305 views
Sliver triangle
Reading through geometric algorithms and code, I've encountered a term I'm not familiar with, and even the mighty google has not been that helpful:
What is a sliver triangle ?
From what i ...
7
votes
4answers
268 views
Maximum area of a triangle
I have been attempting to solve the problem here which is:
Given three concentric circles of radii 1, 2, and 3, respectively, find the maximum area of a triangle that has one vertex on each of ...
7
votes
4answers
226 views
Right triangle where the perimeter = area*k
I was doodling on some piece of paper a problem that sprung into my mind. After a few minutes of resultless tries, I advanced to try to solve the problem using computer based means.
The problem ...
7
votes
3answers
197 views
Elementary Geometry
The side of the square measures $1\ \mathrm{cm}$ , and $AC = 1\ \mathrm{cm}$, find the value of $AB$
7
votes
1answer
77 views
Geometric inequality with a triangle
The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
6
votes
2answers
637 views
Is an equilateral triangle the same as an equiangular triangle, in any geometry?
I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry?
I know they are the same in ...
6
votes
2answers
169 views
Similarity of Triangle problem
Given: AD & PS are medians in ΔABC and ΔPQR respectively,
$$\frac{AB}{PQ}=\frac{AD}{PS}=\frac{AC}{PR}$$
To Prove: ΔABC ~ ΔPQR
Figure:
Problem:
In ΔABD & ΔPQS or in ΔADC & ΔPSR or ...
6
votes
1answer
748 views
RHS Congruency test - What makes 90 degrees different?
RHS is a well known test for determining the congruency of triangles. It is easy enough to prove it works, simply use Pythagorus' theorem to reduce to SSS. I thought that it seems strange that this ...
6
votes
2answers
99 views
Concurrency of A'L, B'M, C'N.
Need some help with the following problem.
Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...
6
votes
2answers
122 views
A question on elementary plane geometry
Given a triangle $ABC$, let $S$ be an inner point of this triangle. Let $P$, $Q$, $R$ be the orthogonal projection of $S$ respectively on the three sides of this triangle. Are there beautiful methods ...
5
votes
2answers
242 views
Why is $d\theta/dx$ necessarily $\cos \theta$ in this physics problem? Or am I wrong?
I'm asking this on the math stack exchange because it seems that the key part of this physics problem I'm asking for help on is more related to the geometry of it than the physics of it.
I'm ...
5
votes
2answers
155 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
137 views
How to know location of a point?
I have a circle formed with three given points. How can i know whether another given point is inside the circle formed by previous three points. Is it determinant i need to calculate? Then what are ...
5
votes
3answers
164 views
On Ceva's Theorem?
The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$
$$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$
is usually proven using the property that the area of a triangle of ...
5
votes
1answer
237 views
Geometry Proving Isosceles Triangle
This question seems tricky and I frankly don't know how to start. I will be grateful if someone can provide a solution.
We have a triangle $ABC$ and there is a point $F$ on $BC$ such that $AF$ ...
5
votes
2answers
229 views
Why Doesn't This Integral $\int \frac{\sqrt{x^2 - 9}}{x^2} \ dx$ Work?
I am trying to solve this integral, which is incorrect compared to Wolfram|Alpha. Why doesn't my method work?
Find $\int \frac{\sqrt{x^2 - 9}}{x^2} \ dx$
Side work:
...
5
votes
3answers
512 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 ...
5
votes
1answer
32 views
Sum of medians of a triangle
I'm very confused because I don't know how I can prove that the sum of the medians of a triangle is equal to the vector zero. Can someone give me a tip or something? Thanks! (And sorry if this ...
5
votes
3answers
76 views
What characteristic of the triangle leads the the existence of the orthocenter
We all know that all three altitudes of a triangle meets in the orthocenter of the triangle. It's a quite classical problem and is proven.
However, what I really wanna know is what characteristic of ...
5
votes
2answers
2k views
Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry
I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
4
votes
3answers
2k 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?
4
votes
4answers
449 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
4answers
75 views
How is this angle relation true?
Either I'm silly and I'm missing something very simple, or my text book is incorrect. I'm trying to verify a line in the text book which claims that sin(a) = s/r. I can't seem to prove this to myself ...
4
votes
3answers
404 views
Why is the inradius of any triangle at most half its circumradius?
Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer.
I know of two proofs of this fact.
Proof ...
4
votes
2answers
111 views
Showing whether two numbers are equal or not
$\dfrac{\sin (2x+y)}{\sin (2x)} =\dfrac{\sin (x+2y)}{\sin (2y)}$,where $0<x,y\le\dfrac{\pi}{4}$ .
Can I show that $x=y $ or find two numbers $x,y$ such that $x\not=y$?
4
votes
1answer
112 views
Find 2nd leg of right triangle with known leg and perimeter
This is probably a very basic question for this site, but it got me stumped.
For a right triangle with one leg (A) and perimeter (L) given, how do I calculate the hypotenuse (C) and second leg (B)? I ...
4
votes
2answers
279 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
4answers
143 views
Is every prime number the leg of exactly one right triangle with integer sides? What's wrong with my argument that this is impossible?
The problem is: "prove that every prime number is the leg of exactly one right triangle with integer sides." However, I seem to have proved that this is impossible. What did I do wrong here?
Let ...
4
votes
1answer
537 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 ...
4
votes
3answers
255 views
geometry triangles side-side-side | prove my teacher she is wrong?
First time I'm here, I'M REALLY frustrated by now.
So I'll just give u the question first.
...
4
votes
2answers
20k views
Solving Triangles (finding missing sides/angles given 3 sides/angles)
What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
4
votes
2answers
101 views
Equilateral triangle geometric problem
I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle $P$. The distance $|AP|=3$ cm, $|BP|=4$ cm, $|CP|=5$ cm.
It is the red ...
4
votes
1answer
158 views
Similar - perspective triangles implies corresponding sides are parallel?
In a general homothetic transformation, if two triangles have corresponding sides parallel then the lines joining respective vertices are concurrent at the homothetic center. I was wondering if the ...
4
votes
2answers
67 views
Coordinates of parallel triangle with a distance of 'd' between the parallel edges?
I have a triangle with Co-ordinates $\{(x_1,y_1),(x_2,y_2),(x_3,y_3)\}$. I need to find co-ordinates of a triangle,whose edges are exactly $\alpha$ distance from previous triangle. Below is the figure ...
4
votes
2answers
78 views
Minimizing the length of a pipeline between cities
I have been trying to minimize piping going to two different cities. City A is located at $(0,4)$ and city B is located at $(6,3)$. The cities must connect to the $x$-axis (the main pipe line.) It ...
4
votes
2answers
196 views
Combinatorics. Inscribed Triangle in a decagon. No shared sides.
How many different triangles can be inscribed inside a regular decagon such that
the triangle shares its vertices with the vertices of the decagon, but the triangle shares none of its sides?
Here is ...
4
votes
2answers
697 views
Can every triangle be divided into five isosceles triangles?
Moderator Note: this is a question from the Federal Mathematics Competition 2013.
That's my problem: Can every triangle be divided into five isosceles triangles?
I've got to give evidence why ...
4
votes
1answer
457 views
Whats the sum of the length of all the sides of a triangle?
You are given triangles with integer sides and one angle fixed at 120 degrees. If the length of the longest side is 28 and product of the remaining to sides is 240, what is the sum of all sides of the ...
4
votes
1answer
38 views
How to find area of triangle from its medians
The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is
a) $48$
b) $144$
c) $24$
d) $72$
I don't want whole solution just give me the hint how ...
4
votes
1answer
71 views
Packing three squares into an equilateral triangle
I am trying to pack 3 equal, largest possible sized squares into an equilateral triangle.
4
votes
1answer
100 views
$C^2=A^2+B^2-2AB \cdot\cos(c)$ getting a different answer than creating a third triangle with the distance formula?
I have the following triangle:
The side going up has a length of 96, the side going down has a length of 112. The angle closest to the center is 91 degrees broken up into 62 and 29 degrees from ...
