For questions about properties and applications of triangles

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1answer
49 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.
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2answers
47 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 $(AB), (BC), (CA)$ ...
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1answer
48 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?
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2answers
51 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. ...
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2answers
45 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!
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3answers
257 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 ...
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1answer
51 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: $OH^2=9R^2-(a^2+b^2+c^2)$. ...
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0answers
68 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 ...
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1answer
89 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 ...
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1answer
44 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 ...
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1answer
73 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, ...
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0answers
54 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 ...
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1answer
20 views

How to find last point of this triangle triangle?

How to find last point of this triangle triangle? I got $(1,7)$ and $(0.5, 4)$. The equations are $y = 3|2x − 1| + 4$ and $y = −|x − 4| + 10$
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0answers
44 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 ...
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4answers
216 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 $80$. ...
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1answer
97 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
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1answer
29 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 ...
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1answer
43 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 ...
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2answers
57 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 ...
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1answer
38 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 ...
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1answer
179 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
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1answer
47 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 ...
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0answers
44 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 ...
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2answers
38 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.
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1answer
235 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 ...
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1answer
75 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 ...
2
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1answer
35 views

Inequality relating to product of sides of convex quadrilateral

We have been that length of both the diagonals are equal to $x$. What can be the maximum value of the product of length of sides? It is obvious that an upper bound exists, but I can't get the ...
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0answers
50 views

Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
0
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1answer
28 views

calculate the angles of a triangle?

If we have a triangle $ABC$ and we only know three things: -The angle $A$ -The length $AB$ -The length $AC$ Is it possible to calculate the other angles: $B$, $C$ All what I can think of ...
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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 ...
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2answers
76 views

No. of equilateral triangles required to completely fill a bigger equilateral triangle

$\triangle ABC$ is equilateral with side length=2.1cm Smaller equilateral triangles with side length=1cm are placed over $\triangle ABC$ so that it is fully covered. Find the minimum number of such ...
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1answer
149 views

Formula to calculate a side of triangle with given angle

I have triangle like in the picture. The known angles: α (total angle of the I-J-K2 triangle) b (total angle of the I-P2-K2 and I-P1-K2 triangles) The known 3D points with X,Y,Z-coordinates: ...
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1answer
283 views

Congruency in bow-tie triangles

We've just started congruency in my class, and we've stumbled across a question which goes like this: Prove that ∆AOB $\equiv$ ∆COD My teacher told us that the way to solve this is using the ...
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2answers
270 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
112 views

Construct an equilateral triangle given a line segment

Given two vertices that make up a line segment (x1,y1) and (x2,y2), how can we find the third vertice that would make up an equilateral triangle? I'm looking to derive the third vertex algebraically, ...
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2answers
56 views

Equilateral triange, sum…

Just a short question: In a triangle we have $\sum (\frac{a}{b+c})^24. Is the triangle equilateral? I have derived
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2answers
273 views

What is the history of the use of the term “scalene triangle”?

A "scalene triangle" is a triangle with three unequal sides. As far as I can tell, this term is not in much use in serious mathematics — in fact, before I became a high school math teacher, I'd ...
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1answer
37 views

How to determine the range of a angle measure?

In $\Delta$ $KLM$, $KL=20$ $LM=13$ m$\angle K$$=40$. What is the range for angle $M$'s measure? Something like between $90^{\circ}$ and $180^{\circ}$
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1answer
50 views

Year 10 - Trigonometry

Please ignore the pencilled 4m in the diagram but I really need to know what the length of the bottom line - line DC - is. A procedure or tips on how to calculate this would be useful. Also, is the ...
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1answer
136 views

If this relation holds, then is the triangle equilateral?

Let $ABC$ be a triangle. If $$\sum_{cyc}\frac{BC}{4AC\cos^2({\frac{\angle BAC}{2})}+BC}=\frac{3}{4}$$ then the triangle is equilateral? We can check if we set $\widehat{BAC}=\pi/3$ and $AB=BC=CA$ that ...
0
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1answer
31 views

New Angle When Opposite Side is Halved

Suppose you have a right triangle with any length sides. The value of one of the angles is $\theta$ and the opposite side is a. If I change the triangle so that the new length of side a is $\frac a2$, ...
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0answers
78 views

Prove that the maximum volume of a triangular-base prism is $\sqrt{\dfrac{K^3}{54}}$ where K is the area of three triangles containing a vertex A

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and ...
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1answer
127 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
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1answer
54 views

Find a right angle triangle in with 3 vertices and one parameter

Given three coordinates, which could be $A=(7,3)$, $B=(2,4)$, $C=(k,-2)$ I want to find the values of $k$ that make a right angle diagram out of the three points. So I initially was thinking to find ...
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1answer
38 views

An Inverse Cosine Problem

Here is my problem: $$ \sin(\cos^{-1} \frac{2}{5} ) $$ I know how to do it for the most part; I just draw a triangle with sides 2,5 and √21 and I then find the sine (opposite/hypotenuse) of the ...
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1answer
59 views

Finding the largest angle of a triangle

The sides of a triangle are $(x^2+x+1), (2x+1)$ and $(x^2-1)$. Then what is the largest of the 3 angles of triangle?
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1answer
44 views

Sin and Cos relationship with Triangle sides

In a triangle ABC, ${sinA < \frac{a}{c}}$ and ${cosA > \frac{b}{c}}$. Which of the statements below are always false regarding triangle ABC? ABC is an acute triangle ABC is an isosceles ...
0
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1answer
68 views

Squares constructed externally on the sides of a triangle and concurrent lines

On the sides $BC, CA$ and $AB$ of the triangle $ABC$ we construct externally the squares $BCDE, ACFG $ and $ABHI$. Denote $A', B'$ and $C'$ the intersectiond points of the lines $BF$ and $CH$, $AD$ ...
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5answers
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Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
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2answers
48 views

What is the nature of Triangle if AB/AC=1/2 angle (BAC)=60°

What is the nature of Triangle if $\frac{AB}{AC}=\frac12$ and $\angle BAC=60^{\circ}$?. Can we use ratio between side lengths?