Tagged Questions

For questions about properties and applications of triangles

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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 ...
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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}{...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $180^\circ$ ?
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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 ...
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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 ...
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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 ...
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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 $T$?
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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: $$\frac{\overline{AB}+\overline{BC}+\overline{CA}}{2}<\overline{AO}+\overline{BO}+\overline{CO}<\overline{AB}+\overline{BC}+\overline{CA}$$ ...
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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?
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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 ...
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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 ...
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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 ...
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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: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ... 2answers 1k views How to derive the law of cosines without the pythagorean theorem To me, it seems that the Pythagorean theorem is a special case of the law of cosines. However, all derivations that I can find seem to use the Pythagorean theorem in the derivation. Are there any ... 3answers 2k views Equilateral triangle geometric problem I have an Equilateral triangle with unknown side$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$cm. It is the red ... 3answers 353 views Smallest square containing a given triangle Given a triangle$T$, how can I calculate the smallest square that contains$T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ... 2answers 184 views The concurrence of angle bisector, median, and altitude in an acute triangle$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$BC,CA,AB$... 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$|CA|+|CB|=2|AB|$? 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 $$\alpha a + \beta b + \... 3answers 4k views Proving the length of angle bisector How do I prove that a triangle with sides a, b, c, has an angle bisector (bisecting angle A) is of length:$$\frac{2 \sqrt{bcs(s-a)}}{b+c}$$I have tried using the sine and cosine rule but have ... 2answers 60 views Finding angle in an equilateral triangular pyramid Given an equilateral triangular pyramid (refer the below diagram) \Delta ABC & P is any point inside the triangle such that {PA}^{2}={PB}^{2}+{PC}^{2}, then \angle BPC is - I am unable ... 4answers 197 views Recurrence relation for right-angled triangles stuck-together Given the image: and that x_0 = 1, y_0=0 and \text{angles} \space θ_i , i = 1, 2, 3, · · · can be arbitrarily picked. How can I derive a recurrence relationship for x_{n+1} and x_n? I ... 4answers 95 views find the measure of AMC if M is the midpoint of BC then find the measure of AMC. I tried to use the angles to find AMC but I don't know how to use that M is the midpoint of BC. 1answer 144 views How to prove the property of the Lemoine point of a triangle? From Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How to ... 3answers 892 views Heronian triangle Generator I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm$$a=n(m^{2}+k^{2})b=m(n^{2}+k^{2})... 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 ... 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: $$\... 1answer 143 views Prove the inequality \frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3} Let a, b, c be the three side lengths of a triangle. Prove that$$\frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\geq 3$$Under what conditions is equality obtained? 1answer 325 views Series for envelope of triangle area bisectors The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is$$\... 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$...
For any triangle with sides a,b,c $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$$ I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to ...