For questions about properties and applications of triangles

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14
votes
3answers
5k 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 ...
9
votes
9answers
5k 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 B}{...
6
votes
5answers
226 views

Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...
0
votes
1answer
75 views

Triangle Formula for alternative Points

With this formula I tried to calculate the third point in a triangle: http://math.stackexchange.com/a/1553606/126977 I know the length between the two point points and the angles at this corners: ...
9
votes
5answers
16k views

Is there any equation for triangle?

Like there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x$ ...
7
votes
3answers
2k 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 1:...
187
votes
7answers
25k 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 ...
7
votes
3answers
3k views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
0
votes
1answer
4k views

probability of three random points inside a circle forming a right angle triangle

three points are randomly chosen on a circle. what the probability that 1.triangle formed is right angled triangle. 2.triangle formed is acute angled triangle. 3.triangle formed is obtuse angled ...
13
votes
4answers
1k 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 ...
2
votes
3answers
417 views

Vector path length of a hypotenuse

Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to: $ P_n = P_x + P_y = \frac{n}{2}\times w + \frac{n}{2}\...
1
vote
2answers
1k views

Convert Equilateral triangle to Isosceles triangle

Let an equilateral triangle have the length of each side an integer $N$. I need to find if it is possible to transform the triangle keeping two sides fixed and alter the third side such that it still ...
62
votes
10answers
17k 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 ...
22
votes
7answers
63k 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 ...
6
votes
2answers
2k views

Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
5
votes
1answer
1k views

Equality of triangle inequality in complex numbers

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
43
votes
14answers
8k 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 ...
5
votes
1answer
3k views

Find the point in a triangle minimizing the sum of distances to the vertices

Given a triangle in a plane with vertices A, B, C, find the point T that minimizes the sum of distances between A-T, B-T, and C-T. I can experimentally determine this point by sampling the space and ...
3
votes
4answers
331 views

Triangle Inequality with Complex Numbers

I was wondering how to prove the triangle inequality with complex numbers: Verify that the function $d(z_1, z_2)$ is a distance funtion on $\mathbb{C}$ and also on any subdomain on $\mathbb{C}$. I ...
2
votes
2answers
12k views

How to find the third coordinate of a right triangle given 2 coordinates and lengths of each side

p2 |\ |b\ | \ A| \C | \ |c___a\ p1 B p3 If given point p1 & p2, side A & B how would you find point p3? I know given this information you ...
2
votes
1answer
462 views

Sum of Angles in a Triangle.

Can anyone please explain how to form a better idea in understanding sum of measures of angles in a triangle is $180^\circ$ ?
1
vote
2answers
44 views

Prove that the triangles $ABC$ and $AB^{'}C^{'}$ have the same incentre.

The question is as follows if $ABC$ is a triangle, with $AD$ as the internal angle bisector of $\angle A$ with $D$ at $BC$ and $B^{'}, C^{'}$ are reflection of points $B$ and $C$ in $AD$. Show that ...
1
vote
1answer
2k views

Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
3
votes
1answer
3k views

calculating the Fermat point of a triangle

Is there any algorithm by which one can calculate the fermat's point for a set of 3 points in a triangle? a fermat's point is such a point that the sum of distances of the vertices of the triangle to ...
3
votes
5answers
301 views

Probability distribution for the perimeter and area of triangle with fixed circumscribed radius

Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
1
vote
1answer
402 views

Prove sum of distance from triangle vertices to a point inside triangle is more than semiperimeter and less than perimeter

If $O$ is a point inside $\triangle ABC$,Prove: $$\frac{\overline{AB}+\overline{BC}+\overline{CA}}{2}<\overline{AO}+\overline{BO}+\overline{CO}<\overline{AB}+\overline{BC}+\overline{CA}$$ ...
17
votes
1answer
119k 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?
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 ...
20
votes
4answers
3k 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 ...
16
votes
4answers
2k 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 ...
8
votes
4answers
285 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
8
votes
2answers
1k views

How to derive the law of cosines without the pythagorean theorem

To me, it seems that the Pythagorean theorem is a special case of the law of cosines. However, all derivations that I can find seem to use the Pythagorean theorem in the derivation. Are there any ...
7
votes
3answers
2k 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
3answers
353 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
4
votes
2answers
184 views

The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
2
votes
2answers
181 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
5
votes
1answer
97 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + \...
3
votes
3answers
4k 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 ...
3
votes
2answers
60 views

Finding angle in an equilateral triangular pyramid

Given an equilateral triangular pyramid (refer the below diagram) $\Delta ABC$ & $P$ is any point inside the triangle such that ${PA}^{2}={PB}^{2}+{PC}^{2}$, then $\angle BPC$ is - I am unable ...
3
votes
4answers
197 views

Recurrence relation for right-angled triangles stuck-together

Given the image: and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i , i = 1, 2, 3, · · ·$ can be arbitrarily picked. How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$? I ...
2
votes
4answers
95 views

find the measure of $AMC$

if $M$ is the midpoint of $BC$ then find the measure of $AMC$. I tried to use the angles to find $AMC$ but I don't know how to use that $M$ is the midpoint of $BC$.
1
vote
1answer
144 views

How to prove the property of the Lemoine point of a triangle?

From Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How to ...
4
votes
3answers
892 views

Heronian triangle Generator

I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm $$a=n(m^{2}+k^{2})$$ $$b=m(n^{2}+k^{2})$$ $$...
2
votes
1answer
5k views

Calculating circle radius from two points on circumference (for game movement)

I'm designing a game where objects have to move along a series of waypoints. The object has a speed and a maximum turn rate. When moving between points p1 and p2 it will move in a circular curve ...
0
votes
1answer
75 views

Radius of circumscribed circle of triangle as function of the sides

Given the length ot the sides $a , b$ and $c$ of $ \triangle ABC$. What is the length of the radius of the circumcribed circle? After some formula substitution I came to the monster formula: $$ \...
0
votes
1answer
143 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?
8
votes
1answer
325 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 $$\...
4
votes
1answer
112 views

Prove the triangle is equilateral given that a quadrilateral related to its circumcircle is a kite

Let $\triangle ABC$ be a triangle. Let $Γ$ be its circumcircle, and let $I$ be it’s incenter. Let the internal angle bisectors of $∠A,∠B,∠C$ meet $Γ$ in $A',B',C'$ respectively. Let $B'C'$ intersect $...
3
votes
4answers
133 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
3
votes
3answers
517 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...