For questions about properties and applications of triangles

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11
votes
2answers
3k 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 ...
7
votes
3answers
1k 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 ...
7
votes
5answers
6k 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 $x$ ...
12
votes
4answers
594 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
278 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 + ...
1
vote
2answers
7k 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 ...
55
votes
10answers
12k 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 ...
6
votes
3answers
882 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 ...
0
votes
1answer
3k 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 ...
4
votes
3answers
90 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. Can anyone help? :)
2
votes
3answers
51 views

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point ...
2
votes
1answer
373 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 are 180 degrees.
0
votes
1answer
965 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 $T$?
17
votes
1answer
78k 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
148 views

In triangle, $\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$

To prove $$\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$$ My approach : $$ \begin{align} \text{L.H.S.} & = ...
4
votes
2answers
126 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
172 views

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

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
41
votes
14answers
6k 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 ...
10
votes
3answers
18k 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 given ...
5
votes
1answer
94 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 + ...
2
votes
4answers
158 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 ...
7
votes
4answers
658 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 ...
5
votes
2answers
1k views

Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three ...
4
votes
3answers
596 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
3k 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 ...
3
votes
3answers
304 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 ...
3
votes
4answers
103 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 ...
2
votes
2answers
134 views

When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
2
votes
1answer
2k views

Related rates: Find dA/dt of triangle, given d(theta)/dt — Can't come to textbook answer

The question: ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the ...
2
votes
1answer
2k 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 ...
1
vote
5answers
92 views

Find the length of a leg of a right triangle, given the area and the length of the other leg

The length of one leg of a right triangle is $(x - 6)$ centimeters, and the area is $(\frac12 x^2 - 7x + 24)$ square centimeters. What is the length of the other leg? I think the equation that I need ...
1
vote
1answer
584 views

equality of triangle inequality

$z$ 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 ...
0
votes
2answers
55 views

Linear algebra - find all possible positions of the third corner?

An equilateral triangle lies in the plane $x + y - z = 1$ and corners in points $(1, 1, 1)$ and $(2, 1, 2)$. Determine all possible positions of the third corner?
-2
votes
2answers
161 views

Consider a triangle with sides, $3,4,5$, does $3^2+4^2=5^2$ hold for such a triangle.

Consider a triangle with sides, $3,4,5$, let the angle opposite the greatest side $5$ be $\theta$, does $3^2+4^2=5^2$ hold for such a triangle. Now consider a triangle with sides (1,1,$\sqrt{2}$), let ...
31
votes
6answers
17k 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 ...
12
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 ...
18
votes
2answers
612 views

Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
16
votes
3answers
2k 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
2answers
265 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 ...
16
votes
1answer
221 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 ...
11
votes
2answers
3k 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 ...
8
votes
4answers
239 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 ...
3
votes
1answer
122 views

I need help with this geometry question.

Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to ...
18
votes
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 ...
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?
10
votes
2answers
3k 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 ...
8
votes
2answers
177 views

How to prove that $\frac{r}{R}+1=\cos A+\cos B+\cos C$?

How do we prove that for any triangle this holds: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$ I can use this beautiful identity to prove several geometric inequalities, but I have no idea how to prove the ...
8
votes
2answers
762 views

The incenter and Euler line.

It seems well known that the incenter of a triangle lies on the the Euler line if and only if the triangle is isosceles (or equilateral, but that is trivial). Searching the internet, I could not find ...
4
votes
3answers
226 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 ...
3
votes
1answer
110 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...