For questions about properties and applications of triangles

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172
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7answers
21k views

V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
60
votes
10answers
16k 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 ...
43
votes
14answers
7k views

Do two right triangles with the same long hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
31
votes
6answers
21k 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 ...
28
votes
4answers
3k views

I think I see mysterious lines inside triangles—how to prove their existence?

Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ...
27
votes
4answers
2k 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 ...
25
votes
1answer
596 views

Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
19
votes
7answers
47k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
19
votes
4answers
3k 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 ...
19
votes
1answer
917 views

Are there prime lengths in triangle with all integer sides and heights?

Suppose you have a triangle in which all sides and all heights are integer in length (i.e. triangle with sides 20, 25, 15 has heights 15, 12 and 20). Could it be that at least one of those numbers is ...
18
votes
4answers
251 views

A triangle determinant that is always zero

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
18
votes
2answers
684 views

Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
18
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4answers
2k 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 ...
17
votes
1answer
109k 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?
16
votes
7answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle $ABC$, $AB=AC=2$. Which of the following could be the area of triangle $ABC$? Indicate all possible areas: [A] $0.5$ [B] $1.0$ ...
16
votes
4answers
1k views

The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
16
votes
1answer
271 views

Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under ...
15
votes
3answers
780 views

The Notorious Triangle Problem

I was told this question by a friend, who said that their friend had thought about it on and off for six months without any luck. I have then had it for a while without any luck either. It is in the ...
15
votes
2answers
502 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
13
votes
6answers
49k views

how to calculate area of 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
13
votes
4answers
1k views

Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
13
votes
4answers
854 views

What is so special about triangles?!

Take any random triangle. If we draw internal-angle-bisectors of all its angles, they intersect at the same point. If we draw the perpendicular bisectors of each side (although they aren't ...
13
votes
2answers
27k views

How to find surface normal of a triangle

If I have a triangle with $3$ points $P_1, P_2,$ and $P_3$, each with $x, y,$ and $z$ coordinates, how do I find the surface normal $N$ in $x, y,$ and $z$ such that $$N_x+N_y+N_z = 1$$ I'm looking ...
13
votes
1answer
1k 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 ...
13
votes
2answers
113 views

The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?
12
votes
5answers
3k views

Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
12
votes
2answers
4k views

Finding an angle within an 80-80-20 isosceles triangle

The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it. So ...
12
votes
7answers
684 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
11
votes
3answers
218 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 ...
11
votes
2answers
5k views

Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides a, b, c. The "square" which I will refer to, from now on, has all its four vertices on the sides of the ...
11
votes
2answers
4k 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 ...
11
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3answers
2k views

How to prove $\Delta DEF$ is an equilateral triangle?

$\Delta ABC$ is an equilateral triangle and $AD = BE = CF$. Prove that $\Delta DEF$ is an equilateral triangle.
11
votes
1answer
119 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
10
votes
3answers
3k views

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

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 ...
10
votes
5answers
2k views

Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...
10
votes
5answers
309 views

Prove that $\angle FGH = \angle GDJ$

Let $FGH$ be a triangle with circumcircle $A$ and incircle $B$, the latter with touchpoint $J$ in side $GH$. Let $C$ be a circle tangent to sides $FG$ and $FH$ and to $A$, and let $D$ be the point ...
10
votes
2answers
288 views

A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$

Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that: $$8r+2R\le ...
10
votes
4answers
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 ...
10
votes
5answers
812 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 ...
10
votes
5answers
733 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 ...
10
votes
1answer
236 views

Number of ways to dissect a square into triangles of equal area

Monsky's theorem states that it is impossible to dissect a square into an odd number of triangles of equal area. If $n$ is an even integer, I am interested in the number of ways of dissecting a ...
9
votes
9answers
4k views

Prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$

I want to prove \begin{equation*} \tan A + \tan B + \tan C = \tan A\tan B\tan C \quad\text{when } A+B+C = 180^\circ \end{equation*} We know that \begin{equation*} \tan(A+B) = \frac{\tan A+\tan ...
9
votes
5answers
637 views

How to find triangle vertices given midpoints?

I have a task to find vertices if midpoints are given: $M1(2;1)$, $M2(5;3)$, $M3(3;-4)$. I know one way to solve it through making a system of equations with three variables. My teacher says there is ...
9
votes
1answer
1k 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 ...
9
votes
6answers
2k views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
9
votes
1answer
130 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
9
votes
1answer
323 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
9
votes
2answers
865 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 ...
8
votes
2answers
199 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
8
votes
2answers
3k 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 ...