For questions about properties and applications of triangles

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2
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1answer
49 views

To prove in a triangle: $AD^2=AB\cdot AC- BD\cdot CD$

If $AD$ is an angle bisector of $\triangle ABC$ (with $D\in BC$), then we have to prove that: $$AD^2=AB\cdot AC- BD\cdot CD$$ I have no idea how to do this, can this be proved with simple geometry? ...
0
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1answer
25 views

Lengths of the sides of a triangle: sufficient and necessary condition?

For any three positive scales, $a,b,c$, what is the sufficient and necessary condition such that they can form a triangle? Is $a+c>b,a+b>c,b+c>a$ enough? Thanks!
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0answers
28 views

Circles intersecting at A and B [duplicate]

Question: Two given circles intersect at A and B. A straight line through B meets the circles again at C and D. Prove that CD is greatest when it is parallel to the line joining the centres My ...
4
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3answers
183 views

Trigonometry. Finding the angle alpha

Refer the diagram below : What should be the angle alpha such that the variable x is between 7mm and 7.3mm.
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2answers
35 views

Isosceles triangle and scalene triangle

Question: Given the base and vertical angle of a triangle show that its area is greatest when the triangle is isosceles. My attempt: For isosceles triangle (with base given 2x, and vertical ...
0
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1answer
56 views

Can we find out the area of conical frustum by using triangles?

I have been trying to find out the area of conical frustum by using triangles.
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4answers
975 views

Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
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0answers
22 views

Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that ...
0
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1answer
51 views

applied optimization problem- triangle fence

A farmer is trying to fence off a field on the edge of a river. He has two 1km long sections of fence to use to make a triangular field. The edge by the river does not need fencing, and the fence ...
0
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2answers
58 views

The height of a right triangle with legs $a,b$ is equal to $ab/\sqrt{a^2+b^2}$ [closed]

The height of a right triangle with legs $a,b$ is equal to $ab/\sqrt{a^2+b^2}$ Need help with number ii since it asks for a uncommon way of approaching the problem.
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1answer
25 views

Question - Corresponding parts of congruent triangles

Please answer the question below with these specifications: If the answer is yes write a paragraph proof to show which congruence shortcut utilized. Show all rules of geometry that are applied to ...
4
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1answer
36 views

Is a triangle with two equal angles always isosceles?

An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two (only two) ...
0
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1answer
29 views

Related rates question.

Two sides of a triangle have lengths $\sqrt{21}~m$ and $\sqrt{7}~m$. The angle between them is increasing at a rate of $\dfrac{2}{\sqrt{3}}~rad/sec$. How fast is the altitude of the triangle ...
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3answers
53 views

Geometry question involving triangle

Question: $ABC$ is a right angle triangle at $A$. $AD$ is the altitude through A; E is a point on AC such that $AE=CD$. F is a poibnt on AB such that $AF=BD$. Prove that $BE=CF$. Challenge ...
0
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3answers
49 views

Getting 90 degree coordinate of 2 coordinates that you know

I have 2 coordinates and I need to find the third with a 90 degree angle. How could I do this? ...
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4answers
86 views

Geometry question involving triangles given with picture.

Here's the question: $\overset{\Delta}{ABC}$ is a triangle. $D$ is a point on $[BC]$. $|BD|=4$. $|AD|=|CD|$. $\text m(\widehat{CBA})=\alpha=30^\circ$. $\text ...
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1answer
29 views

How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
1
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1answer
44 views

Area of Triangle

The position vectors of $A$ $B$ and $C$ relative to an origin $O$ are given by $OA=(2,1,3)$ $OB=(0,-1,7)$ and $OC=(2,4,7)$ Part i) Show that angle $BAC= \cos^{-1}(\frac{1}{3})$ Part ii) Using the ...
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0answers
44 views

Find all the triangles satisfying $\cos(A)\cos(B)+\sin(A)\sin(B)\sin(C)=1$ [duplicate]

I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$. In the case where the triangle has a right angle, ...
0
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4answers
97 views

How can I find the lenght of the third side of any triangle

I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how ...
0
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1answer
22 views

Find the edges of a triangle from a vertex

If I have a series of three vertices that make up a triangle, how can I take one of these vertices and find the edges that go from that vertex to the other two vertices?
0
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1answer
53 views

The Sine Law: A Simplified Criterion for the Ambiguous Case?

Here is my suggestion for an issue that doesn't seem to be handled well in any online notes that I have seen. Can anyone give a counter-example? If you are given $a,b,$ and $B$ in $\triangle ABC$ ...
0
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1answer
26 views

Triangle problem about a point

Question: If D is a point on the side AB of ABC, find a point X on BC such that the triangles XAD and CAX are equal in area. My attempt: I don't actually know how do I solve this problem. I ...
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1answer
42 views

Geometry and triangles problem

Question: If D be the mid-point of AB and if the internal bisectors of $\angle ADC$ and $\angle BDC$ meet $AC$ and $BC$ at H and I respectively. Prove that $HI \parallel AB$ My attempt: It is ...
2
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1answer
37 views

How to find the number of right angled triangles with integer sides and inradius 2009 ..

Problem : How to find the number of right angled triangles with integer sides and inradius 2009 Please help on this as I am not getting any clue how to proceed this problem. I know that ...
2
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2answers
34 views

Trigonometry : Find the length of side

Can someone tell me how to calculate the length 'd' from the below figure? It is from Lecture 06 - Optical flow : ...
1
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1answer
42 views

A right triangle's incenter problem by pure geometry..

$ABC$ is a right triangle such that $\angle B= 90^{\circ}$ and $BD$ is the altitude to $AC$. Given that: $I$ is the incenter of $\triangle ABC$, $I_1$ is the incenter of $\triangle ABD$ and $I_2$ ...
0
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2answers
53 views

Intersection of a median of a triangle with another line segment

In triangle ABC, M is the midpoint of |BC| and D is the interior point of |AB|. Point E is the intersection of the sides |AM| and |CD|. Prove that if |AD| = |DE|, then |AB| = |CE|. I know that this ...
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1answer
28 views

How can I find Triangle base length?

How can I find base length of the triangle in attached picture? Can I use mid-segment theorem to find the base? Thanks in advance
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2answers
60 views

Properties of triangles in non-Euclidean geometries

As we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a ...
0
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1answer
18 views

Choosing the angle in rectangular coordinates

Find all possible polar coordinates for the point P that has rectangular coordinates ( -2,2 (3)^(1/2) ). At the end, the equation satisfied by which angle ? How to know it ? The cos angle or the sin ...
0
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1answer
29 views

Find the plane a triangle lies on

I am trying to determine if the plane on which two triangles lie intersects for a collision-detection implementation. Unfortunately, I'm stuck at step one, which is finding the plane on which a ...
0
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3answers
24 views

How can I calculate angles between objects at the sky?

There is a polar coordinate system which represents the sky from an observer. The elevation angle is 0 to 90 degrees which corresponds to horizon to zenith. The azimuth angle is 0 degrees (north) ...
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0answers
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Filling an Obtuse Triangle with Equilateral Triangles or a Pre-Defined Shape

I am creating an obtuse triangle of undetermined proportions and I need to find how to fill it with equilateral triangles or a pre-defined shape that can fill it. Any math I've done has been, and is ...
2
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2answers
67 views

Finding the length of the side of the equilateral triangle

Here, ABCD is a rectangle, and BC = 3 cm. An Equilateral triangle XYZ is inscribed inside the rectangle as shown in the figure where YE = 2 cm. YE is perpendicular to DC. Calculate the length of the ...
0
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1answer
45 views

Calculate point P(x,y) in a circle given a radius and angle degree

I'm doing a program in Java to draw a PieChart based on given value as link below. data for piechart Given that the diameter, radius, angle degree, center point (150,150) and First Point A (150,0) ...
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1answer
22 views

Dealing with negative areas— coordinate geometry

Question: Find the area of a quadrilateral in the Cartesian plane, whose vertices are (-4, 5), (0, 7), (5, -5) and (-4, -2) My solution: [I meant to draw ...
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2answers
43 views

Given the area and perimeter of a triangle, find its coordinates

How can we find the coordinates of a triangle, given its area and perimeter? (We can find any triangle that satisfies the given area and perimeter) I tried to find the lengths of the sides of the ...
0
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1answer
25 views

Find the sum of the lengths of line segments $BD$ and $CE$

sorry for the drawing. From a point $D$ on side $AB$, a line $DE$ is drawn through a point $E$ on side $AC$ such that angle $AED$ is equal to angle $ABC$. If the perimeter of the triangle $ADE$ is ...
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1answer
33 views

If $|\alpha|\leq 1$ and $|\beta|\leq 1$, prove that $|\alpha+\beta|\leq |1+\overline{\alpha}\beta|$

Note $\alpha$ and $\beta$ are complex numbers and $\overline{\alpha}$ is the conjugate of $\alpha$. I've tried using variations of the triangle inequality and I couldn't find anything to work.
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1answer
225 views

Proof of a certain lemma in geometry

In the following article: http://yufeizhao.com/olympiad/geolemmas.pdf in the proof of the fact about the diameter of the incircle on page 2, the author claims that the proof that $BD = CF$ follows ...
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0answers
36 views

How many are there triangles with different rational sides, rational area, bisectrixes and 1 rational median?

I've been searching triangles with all elements being rational numbers. However, I've found somewhere on Internet proof that it's not possible. Then, I was searching triangles with maximal possible ...
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1answer
135 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
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2answers
41 views

Find the Angle BAC

AB,AC,BC and h are known and its a isosceles triangle how to find angle BAC?
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2answers
22 views

Value of the angle in isosceles triangle.

I try to find a way to calculate value of one of the isosceles triangle angles when I have given values of its height h = 200 and base ...
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1answer
37 views

Triangles, flagpoles and heights, oh my!

Here is a math question i got from school: On a horizontal plane, there are two flagpoles. One is 20m, and the other is 10m. There is a wire connected from the top of each flagpole, to the bottom of ...
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1answer
32 views

Is any property of orthocenter related in this question?

While practicing mathematics Olympiad questions , i got the below given question . Though the solution is given , I am not able to bypass certain steps ... Can anyone please explain me why angle KPA ...
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2answers
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CD is height of right-angled triangle ABC, M and N are midpoints of CD and BD: prove AM⊥CN

I was having some troubles proving this: CD is the height that corresponds to the hypotenuse of right-angled triangle ABC. If M and N are midpoints of CD and BD, prove that AM is perpendicular to CN. ...
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1answer
41 views

Distance over Time and the Pythagorean Theroem

I know the Pythagorean thereom for the last part. I am not $100\%$ sure with the other parts. Here is the problem: Marty and Rediat got in a fight. They walked away from each other on seperate paths ...
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0answers
40 views

Issue with a right-angled triangle

The area of the right angle triangle is $18\text{ cm}^2$ and the ratio of its legs is $2:3$. What is the length of the hypotenuse? I assumed the lengths of two sides to be $2x$ and $3x$. I used ...