For questions about triangles

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4
votes
2answers
32 views

formula for number triangles

Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards ...
5
votes
3answers
235 views

Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle

I just got a long expression: $$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$ and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides I really need tips how to ...
2
votes
2answers
58 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
-1
votes
2answers
75 views

Cut A Shape Into Two Triangles

I have this shape: , and I want to put a straight line somewhere through the shape to cut it into two triangles. I know that this is possible, but I don't know how. Any help is appreciated!
6
votes
2answers
78 views

Number of triangles in a graph based on number of edges

Given a graph $G(V,E)$, what is the maximum number of triangles that this graph can have in terms of $|E|$? I know that there is a triangle listing algorithm that lists all the triangles in ...
0
votes
4answers
66 views

Calculate the angles of a isosceles triangle

In the triangle below, is there a way to calculate the $x$ and $y$? To be more specific, $b = 12.8\rm\,cm\ $ and $h = 10\rm\,cm$, hence $a = 11.87\rm\,cm$. I don't know what to do from here.
1
vote
1answer
42 views

A geometric inequality

Let $M$ be a point inside the triangle $ABC$. $AM$ intersects the circumcircle of $MBC$ for the second time at $D$. Analogously define $E,F$. Prove the following : $$ ...
1
vote
1answer
27 views

$PC+PD$ is least when the angles $CPA$ and $DPB$ are equal

$C$ and $D$ are two points in the $same$ side if a straight line $AB$ and $P$ is any point in $AB$. Show that $PC+PD$ is least when the angles $CPA$ and $DPB$ are equal No idea how to solve this ...
0
votes
1answer
58 views

Length of a segment on right triangles that share same hypotenuse

I have two right triangles that share the same hypotenuse. Can the length of Xb be found using just the other lengths shown (X, L, Y)? I have only been able to find it by using a combination of the ...
1
vote
0answers
45 views

Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
2
votes
1answer
41 views

Circle theorem/triange angle question

I am doing practise papers and there is one question I cannot understand even with the mark scheme. I have added the pictures below: Question (with added annotations): Mark scheme: The question ...
1
vote
1answer
40 views

Is the given triangle unique?

I was reading Polya's How to Solve It when I came across the following problem. Construct a triangle with an angle, the length of altitude through that angle and the perimeter of the triangle given. I ...
2
votes
1answer
48 views

Hyperbolic Triangles and Uniform thinness

My textbook states that all triangles in hyperbolic space are uniformly thin in the following way: If $ABC$ is a triangle and $x$ is a point on one side, then there exists a point $y$ on one of the ...
0
votes
2answers
43 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. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
5
votes
2answers
210 views

Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
0
votes
3answers
106 views

find angle sine knowing all sides

I know all the sides of an arbitrary triangle but not the angles, and I want to find the sine of any angle. ...
1
vote
3answers
37 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
0
votes
1answer
49 views

To prove inequality for two similar triangles $ABC$ and $A_1B_1C_1$ given that $A_1B_1C_1$ is inscribed in $ABC$

Consider a triangle $ABC$. A directly similar triangle $A_1B_1C_1$ is inscribed in the triangle $ABC$ such that $A_1,\;B_1\;,C_1$ are the interior points of the sides $AC,\;AB\;and\;BC$ respectively. ...
1
vote
1answer
29 views

Find Coordinates on a track

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost ...
0
votes
2answers
41 views

Finding coordinates on a circle

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree ...
0
votes
1answer
36 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
0
votes
1answer
24 views

Finding the ratio of a dissected isosceles?

I have trouble trying to find relationship between sub-triangles.
0
votes
3answers
40 views

Find the number of positive integers $b$

Let $a, b$ be positive real numbers such that $10 < a < b$. Then find the number of positive integers $b$ such that (i) $10, a, b$ are in geometric progression, and (ii) $10, a, b$ form the ...
0
votes
1answer
44 views

Proving by using inequality of triangle

suppose that points a and b are from different sides of a line m. Find a point y on line m such that the absolute difference of the YA and YB is maximal. Show proof.
2
votes
2answers
41 views

Sum of areas are equal

Given an equilateral triangle $(ABC)$ and let $P$ be an arbitrary point inside this triangle. Moreover let $V,W,T$ be the orthogonal projections of the point $P$ on to the sides $(AB), (BC), (CA)$ ...
0
votes
1answer
41 views

Length of sides and type of triangle [closed]

If I have the length of three sides, how do I figure out if it's a right triangle? So what is the formula that will help me find this out?
0
votes
2answers
33 views

Drawing a triangle with 2 known corners and all side lengths

Assume that there are three points $A$, $B$ and $C$. All the pairwise distances are known $(|AB|, |AC|, |BC|)$. But none of the coordinates are known. I want to draw a triangle using those points. ...
0
votes
2answers
42 views

Geometry basic problem

Hy! If i have a triangle with given: b-c=3 cm, a=6 cm and alpha is 30°, how do I draw this? Please help me by telling me where I can find this type of exercises online with explanations. Thank you!
4
votes
3answers
182 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 ...
1
vote
1answer
40 views

How to prove these triangle relations?

$O$ is the circumcenter of triangle $ABC$, whereas $G$ is the centroid and $H$ is the orthocenter. $R$ denotes the circumradius. How can I prove the following relations: $OH^2=9R^2-(a^2+b^2+c^2)$. ...
0
votes
0answers
36 views

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

If i have a right triangle ABC with B being the right triangle 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 ...
3
votes
1answer
42 views

Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which ...
0
votes
1answer
36 views

Number Triangle pattern

I have a number triangle as follows: $$\begin{array}{|c|c|c|} \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ \hline 0 ...
1
vote
1answer
53 views

How to solve this geometry question?

Let 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, CA, AB respectively; and $I_1, ...
1
vote
0answers
32 views

Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...
0
votes
1answer
15 views

How to find last pt of triangle

How to find last pt of triangle. I got (1,7) and (0.5, 4). The equations are y = 3|2x − 1| + 4 and y = −|x − 4| + 10
0
votes
0answers
36 views

Complex Number and Geometry

Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at ...
1
vote
4answers
80 views

A triangle has to find its third side.

Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
1
vote
1answer
39 views

Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
0
votes
1answer
19 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
0
votes
1answer
30 views

How can solve a triangle knowing its area , one side and an opposite angle

I need to calculate the missing elements of a triangle knowing its area one side and the angle opposite the given side. The triangle is not a right angle triangle, nor is it equilateral or isosceles ...
2
votes
2answers
40 views

Side of triangle problem

In triangle $ABC$, $AB=BC=12$. Side $AC$ extended through $C$ a length equal to itself to a point $D$. Point $E$ is on $AB$; $DE$ intersects $BC$ at $F$ and $BF$ equal to 8. Find $AE$ without using ...
0
votes
0answers
19 views

Volume of a Part of a Triangular Prism Enclosed in a Sphere

I'm having trouble finding the volume of the shaded prism. I know how to calculate the volume by extending the height of this prism to create a triangular based pyramid, but I cannot get the same ...
0
votes
1answer
30 views

Show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle CAB$

Let A, B, and C be three points on a circle of radius 1. Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle ...
1
vote
1answer
89 views

Finding the missing coordinate of a point within a 3D triangle

We have an equilateral triangle $ABC$ in 3-dimensional space. The points are known, such as: $A = (x_1,y_1,z_1)$ $B = (x_2,y_2,z_2)$ $C = (x_3,y_3,z_3)$ Point $P$ is on triangle $ABC$. If I know ...
4
votes
1answer
44 views

$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$ in a triangle?

How can I prove that ( $\small{\sum}$ denotes cyclic sum here), for any triangle $ABC$: $$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$$ I don't see where to begin even. Any hints would be ...
1
vote
0answers
19 views

How to find the length of the union of Isosceles triangles

I am given N number of right angles triangles all of which are also Isosceles triangles. For each triangle, I am told where they start on a number line and where they end on a number line with end ...
2
votes
2answers
33 views

Nature of the $\triangle$

In $\triangle$ ABC, the $\angle BAC$ is a root of the equation $3^{1\over2} \cos x + \sin x = {1\over2}.$ Then what kind of triangle is the $\triangle$ ABC.
3
votes
1answer
86 views

Circumcircle of an isosceles triangle and length relation

I was asked to prove the following problem. Consider the following diagram where a triangle $ABC$ lies inside its circumcircle, $D$ is the point where the angle bisector $\alpha$ of $B$ intersects ...
3
votes
1answer
63 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 ...