For questions about properties and applications of triangles

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Triangle inscribed in another triangle

If $a,b,c$ are the sides of a triangle,$\lambda a,\lambda b,\lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides $a$ and $\lambda c$,prove that ...
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1answer
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orthocentre and triangle related question

$AD$, $BE$, and $CF$ are the altitudes of triangle $ABC$ with orthocentre $H$, then $C$ is the orthocentre of which triangle? Answer: triangle $ABH$. Please explain.
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2answers
54 views

Does the centroid of a triangle ever fall outside of its Morley's triangle?

Let $T$ be a triangle, and $M$ its (first) Morley triangle:                     (Image from Bruce Shawyer web page.) Q1. Does the ...
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4answers
109 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
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1answer
353 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
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1answer
84 views

Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$

Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot ...
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2answers
11k views

Can we find the perimeter of a triangle given only its base and height?

How do you find the perimeter of a triangle that you only have the base and height measurements for? Please use ENGLISH and not difficult mathematical formulas! I understand little algebra so avoid ...
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1answer
61 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
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3answers
181 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
4
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1answer
86 views

How to prove $\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$?

For any triangle $ABC$, prove that: $$\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$$ I have tried many approaches but none seems to work. I noted that $\cos(\frac{B-C}2)=\frac{AM}{2R}$, where $M$ is ...
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2answers
278 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 ...
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2answers
279 views

Geometric inequality: $2r^2+8Rr \leq \frac{a^2+b^2+c^2}{2}$

Suppose $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $R$ and $r$ are its circumradius and inradius respectively. How can one prove the following inequality? $$2r^2+8Rr \leq ...
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2answers
110 views

Proof of a geometric statement

If $D$ is a point inside a triangle $\triangle ABC$ then how the following statement is true. statement: $AB+AC>BD+DC$. I have tried in the following way but it seems to me defective. ...
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6answers
560 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 ...
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2answers
97 views

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

Given ∆$ABC$ and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$ When does equality holds?
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1answer
1k views

Find coordinates of vertex in right triangle

I have a right triangle with known points $A(x_1,y_1), B(x_2,y_2)$ and known cathetus $AC$ and $BA$. I need to find the coordinates of point $C$.
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2answers
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Interesting problem in congruence of triangles

While solving the exercises of my book I came across this interesting problem: $\triangle ABC$ is isosceles triangle with $AB=AC$. D is a point on base BC such that $AD$ perpendicular on $BC$. To ...
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1answer
564 views

Calculate 3rd point of a triangle, given 2 points and all angles in 2D

I have stumbled upon an interesting problem. I tried to find an answer here but there are just too many similar threads which did not really help me, so I was trying to figure it out by myself. The ...
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2answers
59 views

Substitute for finding the hypotenuse of a right triangle?

All of us know the way to calculate the hypotenuse of a right triangle: Using the Pythagorean Theorem. I came up with a substitute to this. Let the shortest leg of the right triangle be '$a$' units, ...
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0answers
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Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would i find the (x,y) coordinates of the triangle.

Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would I: Create a formula for finding the x coordinate Create a formula for finding the y coordinate of the ...
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3answers
28 views

Scalene triangle with semicircles mensuration

I was recently going through a mensuration sum from a tenth grade board exam book. This one particular question stumped me, and I spent the entire evening thinking of this, but to no avail. The ...
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0answers
26 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
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2answers
47 views

What is the symbol to denote that two triangles are similar?

Does there exist a unique symbol to denote that two triangles are similar to each other without resorting to using the phrase "is similar"?
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4answers
25k 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 ...
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36 views

What can be said about triangle with certain condition?

This question comes from 1988 Irish Mathematical Olympiad, for all those interested. A mathematical moron is given the values $b,c,\alpha$ for a triangle $ABC$ and is required to find $a$. He does ...
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3answers
2k views

Proof of Cauchy–Schwarz inequality

I was reading about the Cauchy–Schwarz inequality from Courant, Hilbert - Methods Of Mathematical Physics Vol 1 and I can not understand what they mean when they said the line that has been ...
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0answers
24 views

Given a single point in 3d space, and 3 points that make up a triangle, find the closest point in/on the triangle to the point.

Given point $(p,q,r)$ and 3 points which make up a triangle, find the closest point in the triangle to the point in space. From the triangle, we can find the equation of the plane $Ax+By+Cz+d=0.$ ...
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1answer
304 views

How can I transform a 3D triangle to xy plane

Suppose I am given a triangle ABC and its corresponding vertex coordinates in 3D. I want to transform ABC in such a way so that vertex A lies on global (0,0,0) coordinate, B lies on (dist, 0, 0) ...
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6answers
457 views

New area of triangle if sides are halved

My question is that if we have a triangle, and we divide each of the side by 2 to get a new triangle, what will be the area of the new triangle in context to the original triangle? Please provide a ...
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5answers
134 views

Right Triangles and Altitudes

I am once again stuck on a question about geometry, this problem is about altitudes that crate right triangles: Let there be a triangle that has side lengths of 13, 20, and 21. Given this, find the ...
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0answers
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Finding maximum subset-triangles

To a given base (ab), triangles are constructed by choosing a point (p). How can i find the maximum subset-triangles(*)? (*)subset-triangle: p' is inside the triangle abp. Allowed interceptions from ...
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What is the relationship of these numbers?

I have two problems that closely relate to each other. I am working with angles. When the angle of Y is 90 degrees the answer to the first problem 360 degrees, while answer to the second is 180 ...
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1answer
54 views

How to prove two triangles have the same centroid?

Suppose you have a $\triangle ABC$ and three similar exterior triangles $\triangle BCX$, $\triangle CAY$ and $\triangle ABZ$. How can I prove that the centroids of $\triangle ABC$ and $\triangle XYZ$ ...
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3answers
44 views

If the hypotenuse is $4$ times the height from $A$, prove that one of the angles is $15^\circ$

In a right triangle (with $\angle CAB = 90^\circ$), suppose $|BC| = 4|AD|$ with $AD$ being the height from $A$ to $BC$. Prove that $\angle BCA$ is $15^\circ$. I had a similar problem but with ...
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1answer
2k views

How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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40 views

Create dynamic cities of perspective angle x

I'm creating a tilemap... I found you can create unique building sizes with perspective with six tiles using parallel projection, whose angles are always $45^\circ$ .... this allows you to connect to ...
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2answers
32 views

Find a bisector point of a circle

The coordinates of $A=(x_{0},y_{0}$) and $B=(x_{1},y_{1}$) are given. How to find the coordinates of $C$ and $D$ as per given information below. ABC is equilateral triangle such that $AB=BC=CA$ ...
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1answer
98 views

How show that $ABC$ is equilateral?

Let $D$, $E$ and $F$ be three points on sides $BC$,$AC$ and $AB$ of triangle $ABC$ such that lines $AD$, $BE$ and $CF$ concur at point $M$. If three trianles $MDB$, $MCE$ and $MAF$ have equal areas ...
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1answer
56 views

How to know which side of the right angled triangle is the base?

If we are given a right angled triangle without any angle or length of any side. How we will find that which side is the base, which side is the perpendicular.
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1answer
50 views

Pascal's triangle

I was out sick for a while (2 weeks) and just got back and now we are doing whatever this is! Can someone explain to me what this is or show me a video on how to do it? "Use Pascal's triangle and the ...
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1answer
512 views

How to calculate Fermat point in a triangle most efficiently?

I am aware of this question, but mine is a bit more specific. I want to find the coordinates of the Fermat point for a given triangle. Assuming that no angle in the triangle is larger than 120 ...
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1answer
45 views

Calculate PQ if AC = 20

I need to calculate $PQ$ knowing that $AC = 20$. This is what I got so far: If I call the point between $P$ and $A$, "$M$" and If I call the angle: $$\measuredangle{QPB} = y$$ Then: ...
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1answer
55 views

Sum of inradius of constructed triangle

Let $ABC$ be a triangle with inscribed radius $r$ and circumscribed radius $R$. Let $A′B′C′$ be the triangle for which $A′B′$ is the perpendicular to $OC$ through $C$ and so on. Let $r_1$ be the ...
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2answers
421 views

Find side BC of a triangle given AB, AC, and a relation between $\angle A$ and $\angle B$

A question from my class: In triangle $ABC$, $3\angle A+2\angle B=180$ and $AB=10, AC=4$. So question is, what all can we comment on side $BC$. Can we find its exact length? I have a crude ...
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area of triangle in terms of sides ratio [duplicate]

In $\triangle ABC$, $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively. If $Z$ is on the segment $XY$ such that $\frac{AX}{XC}=\frac{CY}{YB}=\frac{XZ}{ZY}$, prove that the area of ...
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2answers
58 views

Length of a line in an isosceles triangle. (mind boggling )

In an isosceles $\triangle ABC$, side $AB$ and $AC$ are equal in length. There exists a point $D$ on the side $AB$. $\angle BAC$ is $\theta$. The side $AD$ is $2$ units smaller than $AC$. What is ...
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2answers
49 views

Geometry basic problem

If I have a triangle with given: $b-c=3 \space\text{cm}$, $a=6\space \text{cm}$ and $\alpha$ is $30^\circ$, how do I draw this? Please help me by telling me where I can find this type of exercises ...
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0answers
73 views

Moving up the Y axis the length of the hypotenuse of a right triangle

If I have a right $\triangle ABC$ with $B$ being the right angle and length $AB = 50$ and length $BC = 50$. Based on the Cartesian coordinate system if I wanted to move up the Y axis the length of the ...
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1answer
80 views

How to solve this geometry question?

Let $\triangle ABC$ be an acute-angled triangle; $L$, $M$, $N$ be the feet of perpendiculars respectively from $A$, $B$, $C$ to the opposite sides; $D$, $E$, $F$ be the midpoints of the sides $BC$, ...
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31 views

At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...