Questions tagged [triangles]
For questions about properties and applications of triangles.
6,530
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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
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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 ...
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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
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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 ...
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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 ...
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A Trigonometric Inequality in a triangle. [closed]
Prove that in any acute $\Delta ABC,$
$$\sin A \sin B + \sin B \sin C + \sin C \sin A \geq (1 + \sqrt{2\cos A \cos B \cos C})^2.$$
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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, ...
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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 ...
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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:
...
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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 ...
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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 ...
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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 $...
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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 $...
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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 ...
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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 ...
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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} +...
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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
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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 ...
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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^...
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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 ...
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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 ...
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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%...
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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 ...
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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 $...
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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 ...
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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$...
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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 ...
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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 ...
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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 ...
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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)\...
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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.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 $K$....
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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)}{\...
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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 ...
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2
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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 ...
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2
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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 ...
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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 ...
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2
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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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