Questions tagged [triangles]

For questions about properties and applications of triangles.

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Finding the Area of the Largest Isoceles Triangle

Can you please help me find the answer to question (a)? I've been looking for the answer for hours. It's still wrong, please help
user1228091's user avatar
2 votes
3 answers
77 views

Let the triangle $ABC$ be $\angle A = 110°$ and $\angle B = 40°$. We consider a point $E$ . Show that $CA=CE$

QUESTION Let the triangle $ABC$ be $\angle A = 110°$ and $\angle B = 40°$. We consider a point $E$ inside the triangle $ABC$ so that $\angle ECB = 10°$ and $\angle EBC=20°$. Show that $CA=CE$. my ...
Ionela Buciu's user avatar
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0 answers
19 views

Can two triangles not be congruent but five elements in triangle are same

Can two triangles not be congruent but five elements in triangle are same
Adi shankaracharya's user avatar
1 vote
4 answers
120 views

How to show that two certain chords of a circle passing through the incentre of a given triangle are equal?

Let $I$ be the incenter of a triangle $\triangle ABC$. The circle $AIB$ meets the sides $BC$ and $AC$ at points $M$ and $N$, respectively. I'm trying to prove then that $BM=AN$. Here's a figure for ...
math-physicist's user avatar
-2 votes
0 answers
73 views

How were the $x$ and $y$ coordinates of intersecting point calculated?

Right angled triangle at $(0,0) (0,100)$ and $(200,0)$ Came across the solution of this in my coding and the co-ordinates of intersecting point of hypotenuse and line passing through origin and user ...
Abdul Moiz's user avatar
-2 votes
0 answers
21 views

A Trigonometric Inequality in a triangle. [closed]

Prove that in any acute $\Delta ABC,$ $$\sin A \sin B + \sin B \sin C + \sin C \sin A \geq (1 + \sqrt{2\cos A \cos B \cos C})^2.$$
Entrepreneur 's user avatar
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How to find length of side of smaller scalene triangle inside larger scalene triangle

I'm dealing with a real-world problem that can be abstracted into triangles. The knowns are: length of $AB$ length of $CB$ length of $AC$ length of $EF$ $EF$ and $AC$ are parallel In other words, ...
polyrhythm's user avatar
-1 votes
0 answers
43 views

A problem about a right triangle and a circle [closed]

On the hypotenuse AB of right triangle ABC the altitude CH is drawn. The perpendiculars HK and HE are drawn from the point H to the cathetes (K in AC and E in CB). Find the radius of the circle in ...
Тимофей Главицкий's user avatar
0 votes
1 answer
52 views

The area of a triangle using sine

Please tell me why this statement is true for triangles with an obtuse angle, acute angle and the same two sides a and b: ...
Тимофей Главицкий's user avatar
2 votes
2 answers
98 views

Proving two triangles congruent given two congruent sides and a congruent median

The title was a bit too short for me to fit the full details, so here's the scenario I have. Prove that two triangles are congruent if in two triangles, the median from the common vertex and two sides ...
nadelock's user avatar
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0 answers
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The ratio of the volume of a pyramid when it is divided into two halves

Let's assume we have a pyramid with a multi-dimensional triangular cross-section. If the dimension is equal to 2, it becomes a triangle. Now, if we divide the pyramid (or the triangle in two ...
M a m a D's user avatar
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Question about trapezoid [closed]

Good time of day! I have the following question. Let $ABCD$ is a trapezoid with the base $AD$ and $\angle BAD + ∠ADC \neq 120^{\circ}$. Points $A′$ and $B′$ are located symmetrical to points $A$ and $...
UserFed's user avatar
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1 answer
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ANY IDEAS - A problem about the inscribed and circumscribed circles...

PROBLEM We consider the particular triangle $ABC$ and denote by $I$ the center of the inscribed circle. The circumscribed circle of the triangle of the triangle $AIC$ cuts the line $BC$ at the point $...
Ionela Buciu's user avatar
9 votes
1 answer
295 views

A "New" Special Point in a Triangle.

I was playing with the software Geometry Expressions and I was exploring generalizations of special points in triangles (centroid, orthocenters, etc.) when I stumbled upon this construction. J is ...
Shaktyai's user avatar
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1 answer
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Area of non-right angleTriangle is given, rise/run given, no sidelength is known, calculate triangle height.

I have a triangle of $600m^2 $ area, I do not have any lengths of this triangle, I only have the rise/run of both sides that are not sitting on the x-axis, and the area, can I find the height given ...
b077h34d's user avatar
0 votes
0 answers
23 views

Geometric inequality involving incentre and medians [closed]

Let $I$ be the incentre of $\Delta ABC$ and let $m_a, m_b$ and $m_c$ be the lengths of the medians from vertices $A, B$ and $C$ respectively. Prove that $$ \frac{IA^2}{{m_a}^2} + \frac{IB^2}{{m_b}^2} +...
Entrepreneur 's user avatar
-3 votes
0 answers
35 views

To find angles of a triangle, A, B, and C when the corresponding sides are square roots ie √A, √B, and √C [closed]

The three angles of the triangle are known to be A, B and C. The corresponding sides are square roots √A, √B, √C. Find A, B, C
dsm super master's user avatar
2 votes
4 answers
198 views

Relation between AD, BD and BC

In a triangle $ABC$, $\sphericalangle BAC = 100°$ , $AB=AC$. A point $D$ is chosen on the side $AC$ such that $\sphericalangle ABD = \sphericalangle CBD$, prove that $AD+DB=BC$. I tried solving this ...
Techno Highway's user avatar
1 vote
1 answer
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Geometry with complex numbers. Equilateral triangle.

I have to do such a question: Let ABC be a triangle, whose vertices A, B, C correspond to the complex numbers α, β, γ (the origin is not necessarily at one of the vertices), respectively. Let $ ω = e^...
WOWnas's user avatar
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0 answers
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Equilateral triangles 27-nodes cubic Lattice

How many equilateral triangles of any size and orientation can be found in the 27-node cubic lattice? In each equilateral triangle, there must be three nodes exactly at its corners. 27-node cubic ...
Ravi Maurya's user avatar
1 vote
0 answers
78 views

Is the area of a triangle is less than that of its mean triangle of equal area?

Definitions: We generate random triangles using various distribution (uniform, gaussian etc) for $\theta$ to get the polar coordinates of the vertices $𝑟\cos \theta,𝑟\sin \theta$ in a circle of ...
Nilotpal Sinha's user avatar
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0 answers
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Can the Miquel Point lie on the Triangle?

Here is a link to the Wikipedia article regarding the Miquel point Theorem. https://en.m.wikipedia.org/wiki/Miquel%27s_theorem#:~:text=Miquel's%20theorem%20is%20a%20result,points%20on%20its%20adjacent%...
Rocky.Racoon.'s user avatar
2 votes
1 answer
81 views

What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?

The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a equilateral triangle and it also gives the perimeter is $3\sqrt{3}$. For ...
Nilotpal Sinha's user avatar
0 votes
0 answers
19 views

Average side length of a triangle with perimeter $p$

On the one hand, I think that by symmetry the average side of a triangle with given perimeter $p$ is $\frac{p}{3}$. However (and here I'm probably mistaken), if I look at a side of the triangle, say $...
HappyDay's user avatar
  • 846
3 votes
1 answer
67 views

Does the mean area of triangles with equal perimeter $p$ and circumradius $R$ have a local minima at $p=4R$?

Definition: Isoperimetric triangles are triangles which have the same perimeter and the same circumradius. Isoperimetric area curve: The largest perimeter of a triangle that can be inscribed in a ...
Nilotpal Sinha's user avatar
0 votes
2 answers
83 views

Given points $A$, $B$, $C$ and $D$ lying on a circle and lines $BD$ and $AC$ intersecting at $F$, prove lines are parallel

The diagram shows the points $A$, $B$, $C$ and $D$ lying on a circle. $AC$ and $BD$ intersect at point $F$. $EG$ is tangent to the circle at point $C$. $AD$ is produced to meet the tangent at point $E$...
Talha Ahmed's user avatar
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30 views

Find relation between $\alpha$, $\beta$, $\gamma$ in the given figure(involves a circle and triangles).

Source: MAT $2011$ I cannot decide between two of the MCQ options. Both of them seem correct. (a) $\cos\alpha = \sin(\beta+\gamma)$ (b) $\sin\beta = \sin\alpha \sin\gamma$ (a) If I join the centre ...
acelixis's user avatar
  • 187
1 vote
3 answers
120 views

Which parameters can we choose in order to solve this triangle construction issue

This is follow-on of a question asked yesterday, with real work shown under the form of sketches but not understandable. Visibly, the asker isn't used to formulate mathematics with sentences (his/her ...
Jean Marie's user avatar
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4 votes
2 answers
161 views

Find angle in a triangle given angle bisector and altitude without trigonometry.

In the triangle $ABC$, $BM$ is altitude and $E$ is in that segment such that $CE$ is angle bisector. Also, the angle $EAM = 30º$ and the angle $MCB = 20º$. Find the value of $ABM$. My problem with ...
Trobeli's user avatar
  • 3,232
1 vote
3 answers
84 views

Finding the angle $x$ in $\triangle ABC$

What is the angle $x$ in the following diagram. I could solve this problem by applying the Sine Rule for $\triangle ABD$ and $\triangle BDC$. After some simplification I got $$\sin^2 x =\sin(45-x)\...
Etemon's user avatar
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-3 votes
0 answers
52 views

How many unique triangles of the same type can fit in a circle? [closed]

Given a circle of any radius, i want to find how many unique triangles of the same shape can fit in that circle. All the three vertices of every triangle have to be on the perimeter of that circle. ...
Pika-Chu's user avatar
-5 votes
0 answers
28 views

There are 2 points on 2 sides of a triangle, find angle in Triangle [closed]

Given triangle $ABC$ with angle $\angle ABC = 60^\circ$ and $\angle BCA = 100^\circ$. On sides $AB$ and $AC$, there are points $D$ and $E$ respectively such that $\angle EDC = 2\angle BCD = 2\angle ...
Idham Muqoddas's user avatar
1 vote
1 answer
29 views

Solve for the X/Y (Cartesian) coordinates of the remaining vertices of an isosceles triangle with known Angles & Lengths

I'm attempting to program something and I'm reaching the limits of what I can remember about basic geometry (its been like 15 years since I've taken trig). I feel like I have figured out a large ...
Mark Johnson's user avatar
-1 votes
2 answers
98 views

angle Ceva's theorem

In triangle $ABC$, $P$ is point such that $\angle PAB = 42^{\circ}$, $\angle PBA = 54^{\circ}$, $\angle PAC = 6^{\circ}$, $\angle PBC = 12^{\circ}$. Find $ \angle PCB$. I found the $ \angle PCB ...
Snupi's user avatar
  • 129
3 votes
2 answers
99 views

Finding the angle EDB in triangle ABC, where E is the intersection of the angle bisector of C with side AB and D is a point on BC

This was a question I encountered while looking at some weekly math questions my school had hung in front of the department last week: I was unable to solve it, and now that some time has passed, I'd ...
LogicBeDamned's user avatar
1 vote
0 answers
27 views

Equality of Segments in a Corner with Two Tangent Inscribed Circles

The problem Two circles are inscribed in the corner. Points $A$ and $B$ are points of contact of the first circle with the sides of the angle, points $A_{1}$ and $B_{1}$ are points of contact of the ...
curioushuman's user avatar
0 votes
1 answer
28 views

Finding Value $q$ in a Triangle Given: Base $c$, Height $hc$, and Angle $γ$

The objective is to determine the value $q$ for a specific angle $γ$. Given is a triangle with the known values: base length $c$, height $hc$ relative to this base, the angle $γ$ opposite this base. ...
Steffen's user avatar
1 vote
0 answers
48 views

Vertex circle radii of Pythagorean triples

@Blue was kind with his comments on a previous question here. I'd now like to share some new relationships I found using algegra and my favorite formula for generating Pythagorean triples. In this ...
poetasis's user avatar
  • 6,084
3 votes
1 answer
96 views

Proving that no tile can fill both squares and equilateral triangles

Cut up a square into a finite number of identical tiles. Here is one possibility: How do I prove that the tiles could never be rearranged to form an equilateral triangle (with filled interior and no ...
bobuhito's user avatar
  • 791
17 votes
7 answers
2k views

Showing that the centers of two semicircles and a circle inscribed in a quarter circle form a right triangle

The challenge in this image is to determine the radii of the two semicircles and the full circle. Determining the radii of the two semicircles is straightforward; if the radius of the quarter circle ...
user46802's user avatar
  • 179
0 votes
0 answers
34 views

Which formula does it use here to get the curvature?

I'm reading the wikipedia page https://en.m.wikipedia.org/wiki/Menger_curvature It mensions a so-called well-known formula here but doesn't mention its name. I guess it is heron's formula. But again ...
user900476's user avatar
2 votes
1 answer
67 views

What is the formula for the parameters of an ellipse based on the linear transformation of a triangle containing it?

Let: $T_0$ be a $d$-dimensional triangle ($d \ge 2$) whose incenter is the origin and whose sides have known lengths $a_0$, $b_0$, and $c_0$; the corners of $T_0$ are $\vec{A}_0$, $\vec{B}_0$, and $\...
nben's user avatar
  • 294
1 vote
1 answer
81 views

Calculating $ACB = A'CB'$ given that $ABC$ and $A'B'C$ are equal in area

Given is a triangle $ABC$ with incircle center $I$ and side centers of $AC$ and $BC$ being named $M_a$ and $M_b$, respectively. Let the intersection of the lines $M_bI$ and $BC$ be called $B'$ and ...
Created Maths's user avatar
0 votes
0 answers
23 views

Find the Length of Parallel Line Through Trapezoid's Diagonals Intersection with given Bases Lengths

The problem: A straight line is drawn through the point of intersection of the diagonals of a trapezoid, which is parallel to the bases and intersects the sides of the trapezoid at points $M$ and $K$....
curioushuman's user avatar
3 votes
1 answer
132 views

Prove this trigonometry equation in geometry setting

Equilateral triangle $BCD$ is constructed out of $\triangle ABC$. Let $AA^\prime$ be diameter of $(ABC)$. If $\angle BDA^\prime=\alpha$, $\angle CDA^\prime=\beta$, prove that \[\frac{\sin(B+\alpha)}{\...
youthdoo's user avatar
  • 705
0 votes
3 answers
90 views

Geometry: In the $\triangle ABC, AB=8, BC=7, CA=6$. Let $E$ be a point on $BC$.

In the $\triangle ABC, AB=8, BC=7, CA=6$. Let $E$ be a point on $BC$ such that $\angle BAE=3\angle EAC$. Find $\frac{4(AE)^2}{5}$. In my solution I had started with the apollonius's theorem which ...
The Revolution's user avatar
0 votes
2 answers
71 views

Perpendicularity in a given triangle

I was asked to solve the following problem by a friend: Here, $BC$ is a diameter of the circle, $E$ is the midpoint of the $DC$ arc, $F$ is the midpoint of $BD$, $G$ is the intersection of $FE$ with ...
Tassandro Cavalcante Leitão's user avatar
0 votes
2 answers
104 views

Conjecture: An interesting concurrency concerning incircles [closed]

Conjecture: Choose any point $P$ in the interior of the incircle of triangle $ABC$, and from the points of tangency, $D$, $E$, and $F$, draw lines through $P$ intersecting the incircle a second time ...
Hans Humenberger's user avatar
1 vote
1 answer
36 views

Possibility of determining the third side of a right angled triangle using given three parameters:

Suppose there are two arbitrary side lengths of a right angled triangle that are known to us. There are two possible cases here that I can see: Either one of the side lengths given is the length of ...
Aaditya Soni's user avatar
3 votes
2 answers
207 views

Is finding the area of this rectangle impossible?

One of my students gave this problem and I am feeling quite ashamed that I could not find an answer. It asks for the area of the pink rectangle and it says that the triangle ABC is a right angle ...
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