For questions about properties and applications of triangles

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2
votes
1answer
93 views

Find direction, angle or co-ord of unknown vertices using only distance?

My current issue is that I have a triangle, where I know all the line distances as well as an origin coordinate. Is there any way I can then gain the coordinates of the other vertices with this ...
0
votes
1answer
376 views

Find a point on a plane

I have three points to write the equation of a plane: assume $P_1=(x_1,y_1,z_1),P_2=(x_2,y_2,z_2),P_3=(x_3,y_3,z_3)$. I can also write the equation of this plane. I want to obtain the coordinate of ...
2
votes
3answers
411 views

$\sin{\alpha}+\sin{\beta}+\sin{\gamma}>2$ Where $\alpha$, $\beta$ and $\gamma$ are angles from an acute-angled triangle.

The problem is easy to state: Prove that $$\sin{\alpha}+\sin{\beta}+\sin{\gamma}>2$$ Where $\alpha$, $\beta$ and $\gamma$ are angles from an acute-angled triangle. I only managed to turn it into: ...
4
votes
1answer
134 views

What does relative height to the hypothenuse means?

I have to solve the next problem: Given H (relative height to the hypotenuse) and R (radius of the circle inscribed in the triangle) of a rectangle triangle, can you calculate the value of its ...
1
vote
1answer
65 views

Geometry Proof Triangles

Show that if two of the corresponding angles of two triangles are equal then so is the third. Is there a formal way to prove this? I wanted to just say in one sentence that if two angles are the ...
0
votes
1answer
91 views

Determine the exact location of the centroid?

This is my last question for the day! :P Usually I am good at math but I've been sick for over a year and am now finding it hard to concentrate. :P Triangle CDE has vertices C(-2,4), D(6,2), and ...
1
vote
0answers
189 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
1
vote
2answers
80 views

Parallelograms in triangles

if posssible, could you only give me a few theorems in order to assist me in this question. Thankyou in advance! Links to simple websites would also be appreciated. In triangle $ABC$ $F$ is midpoint ...
1
vote
1answer
61 views

Proving similar triangles

In trapezium $ABCD$, $AB$ is parallel to $DC$. The diagonals $AC$ and $BD$ intersect at $X$, and $XY$ is constructed parallel to $AB$, intersecting at $X$, and $XY$ is constructed parallel to $AB$, ...
0
votes
1answer
117 views

Finding the exact area of a trapizium using similar triangles

IN the trapezium ABCD, the diagonals intercept at M. Let AM= a, BM= b, Cm = c and DM = d, and let Angle AMB be $\theta$. a=6 b=4 c=3 d=2 AB=8 DC=4 $\cos(\theta) = -\frac{1}{4}$ and $\sin ...
2
votes
1answer
416 views

Proof involving angle bisector in an arbitrary triangle

In the above figure, AD is a bisector angle A (angle BAC). How do I prove in a triangle ABC of any dimensions that, $AB > BD$ $AC > CD$ Is it also possible to prove that, $AB > AD$ ...
1
vote
1answer
141 views

Constructing an equilateral triangle from an arbitrary triangle by shifting towards an interior point

Suppose $\triangle ABC$ has no angles greater than or equal to $120^{\circ}$ and let $P$ be any arbitrary point inside $\triangle ABC$. Let $\overline{AP}, \overline{BP}, \overline{CP}$ be the line ...
0
votes
1answer
1k views

Proof of ASA , SAS , RHS , SSS congruency theorem

I have tried searching in many places for some good proofs of these theorems but couldn't find them anywhere . Even my math teacher cannot explain it to me and says that these theorems just work. I ...
1
vote
2answers
273 views

How to prove point A belongs to line t?

I'm stuck at trying to prove that any point $A$ will belong to line $t$ if and only if segments $AB=AC$, where $B$ and $C$ are symmetrical points to the line $t$ and $M$ is the midpoint of segment ...
0
votes
1answer
396 views

GRE triangle area question

I dont understand why AE is 1 if AD is 4 and the ratio between CD and AB is 9/3 or 3
8
votes
2answers
195 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
1
vote
1answer
107 views

Right triangle with equal permeter and height - how to find side lengths?

The question: Suppose there is a right triangle with sides $a$ and $b$ and hypotenuse $c$. Its perimeter is the same as its area, and $b = 6$. What are its side lengths? I just cannot figure ...
2
votes
2answers
177 views

Finding value of an angle in a triangle.

I'm solving some practice problems to prepare for a competitive exam . Here is one which I'm trying to do for some time but still haven't found a solution to : "In the given figure , ∠ABC = 2∠ACB and ...
0
votes
2answers
170 views

Trigonometry? Get the “half” of a triangle from hypotenuse and cathetus

I've only got the following parts of a triangle: Line A to B Line B to C And optionally the Line from A to C if needed? I'm trying to get the point X Now the problem is, i've got absolutly no ...
2
votes
1answer
183 views

A problem related to area of triangles.

I'm solving some practice problems to prepare for a competitive exam . Here is one which I'm trying to do for some time but still haven't found a solution to : " In $ΔABC$ , $E$ and $F$ are such that ...
2
votes
0answers
160 views

About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle.

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...
0
votes
4answers
142 views

How would you measure a right triangle with sides of 1 and root 2?

This may be a silly question, but I saw this diagram on wikipedia and was intrigued: https://en.wikipedia.org/wiki/File:Square_root_of_2_triangle.svg How would such a triangle work in real life? ...
3
votes
2answers
344 views

Confusing angle-chasing question

AB = BC = CD = DE = EF = FG = GA Find angle GAB. Please, I want the correct answer. I know how to solve it, but I am getting confused by the number of triangles in it. I am getting different ...
3
votes
3answers
318 views

Proving a point inside a triangle is no further away than the longest side divided by $\sqrt{3}$

Problem: In a triangle $T$ , all the angles are less than 90 degrees, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the ...
15
votes
2answers
459 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 $ABC$ ...
0
votes
2answers
18k views

Length of hypotenuse using one side length and angle

I bet this question has been asked a million times, but I can't find a straight answer. I need to find the length of the hypotenuse in a triangle where I have one side and all the angles. Example: ...
1
vote
3answers
2k views

A problem on a triangle's inradius and circumradius .

I'm trying to solve the following problem : In $△ABC$, $AB = AC, BC = 48$ and inradius $r = 12$. Find the circumradius $R$. Here is a figure that I drew : ( note : it was not given in the question ...
1
vote
1answer
356 views

Circumcenter coordinates for a isosceles triangle

I'm back, wow, twice a day nowadays. I need to calculate circumcenter coordinates (or at least I hope they're called that) at point C for an isosceles triangle (the circle must be such, that created ...
2
votes
1answer
512 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
1
vote
2answers
63 views

relation between inscribed and circumsdribed circle

Let T be the triangle with side lengths $b_1,b_2,b_3$, and $r_{insc}$ and $r_{cir}$ be the radii of the inscribed and circumscribed circles,respectively, I need to show that $$ \frac ...
0
votes
0answers
95 views

Fastest way to check whether the triangle inequality is satisfied

If we are given the lengths of the three sides of a triangle, and we simply add the 2 smallest sides and check to see if the sum is larger than the third side, will this always yield the correct ...
1
vote
7answers
972 views

An alternative proof of 30-60-90 theorem/

A 30-60-90 theorem in Geometry is well known. The theorem states that, in a 30-60-90 right triangle, the side opposite to 30 degree angle is half of the hypotenuse I have a proof that uses ...
5
votes
3answers
453 views

Prime Number in triangle

I had a question here, the measures of the sides of a right triangle (a single unit) can be prime numbers? If they can not, why?! But, if you can, could you help me find an example?
3
votes
1answer
185 views

the ratio of the following two areas

Suppose you have the following triangle $ABC$: with the following properties: $|AB|=4\cdot |AA'|$, $|AC|=4\cdot |CC'|$, $|BC|=4\cdot |BB'|$. I have to find the ratio of the total area of the triangle ...
6
votes
2answers
142 views

Find The range of $r/R$.

Given a triangle $ABC$ with angle $A=90^{\circ}$. Let $M$ be the midpoint of $BC$. If the inradii of the triangles $ABM$ and $ACM$ are $r$ and$\ R$ respectively, then find the range of $\dfrac rR$ .
1
vote
4answers
132 views

How to mathematically define “on the outer side of the triangle”?

Given the coordinates of a triangle's vertexes, I'm trying to find its Fermat point programmatically. In one step of the algorithm that I'm trying to implement, I have to draw equilateral triangles on ...
4
votes
1answer
445 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 ...
3
votes
1answer
248 views

Could someone explain this animated gif to me in mathematical terms?

I understand that the area of the two squares around the right triangle are the total area of the one that is the hypotenuse. Is this just a proof for the Pythagorean theorem or is there some other ...
0
votes
1answer
304 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
0
votes
2answers
87 views

Sides from angles of a triangle

How does one find the side lengths of a right triangle in relation to each other using just the angles? I have all three angles. Is this even possible?
0
votes
2answers
86 views

Proving an extension to nesbitts inequality: $\frac a{b+c}+\frac b{c+a}+\frac c{a+b}\lt 2$

Prove that $$\frac a{b+c}+\frac b{c+a}+\frac c{a+b}\lt 2$$ given that $a,b,c$ are sides of a triangle. I know that the above is $\ge \frac32$ but how will you prove the above? I know this might ...
4
votes
1answer
554 views

find the rate of change of the area of triangle pulled by three people from its sides

this is the problem of my curious mind(I am it's designer!) . three people each having the rope attached by the end of the 3 sides of triangle ABC , pull the triangle with speed U in the direction ...
0
votes
3answers
84 views

Ratio of side length of triangle?

In triangle ABC, we choose a point D at AB such that the length of AD=1/2 AB, and point E at AC such that AE=3EC. F is intersection point of CD and BE. What is the ratio of CF/FD and BF/FE?
1
vote
1answer
855 views

Calculate Triangle Ground using Height and Top Angle

Is it possible to calculate the ground of a triangle only using the height and top angle. Click here to see a poorly draw sketch of what I'm trying to calculate. So is it possible and how, to ...
8
votes
4answers
246 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 ...
-1
votes
2answers
78 views

Is the orthocenter and incenter of a triangle the same point?

Although the orthoceneter and the incenter of a triangle are technically different things: The point in which the three altitudes of a triangle meet is called the orthocenter of the triangle. ...
0
votes
1answer
22 views

How do I find out the coordinates, interpolating across an angled line?

Suppose I know the coordinates of $A$ and $B$. The angle $X$ does not mean the total angle between the red lines, but rather how far along the angle that the purple line is. What is the easiest way ...
0
votes
1answer
146 views

portion of areas of two triangle

let us consider following picture we are given that this two line is parallel and also $AC=1/3 * AD$,we should find portion of areas of $ABC$ and $BCD$,now because $AB$ is one third of $AD$, ...
1
vote
1answer
756 views

A Question related to triangle and centroid .

The following is a geometry question I can't seem to get. "Consider an acute angle △ABC. Points D, E, F are mid points of sides BC, CA and AB respectively. G is the centroid of △ABC. Area of △AFG = ...
1
vote
1answer
393 views

point which minimises the sum of the distances from the sides in a triangle

I want to know the point in the plane for which the sum of the perpendicular distances from the three sides of a general triangle is minimized. Also, is the point unique?