For questions about properties and applications of triangles

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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 ...
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1answer
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Proving identity on triangles

Lower-case letters denote opposite sides and uppercase letters denote corresponding angle
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Proving identity relating to properties of triangles [on hold]

Prove that $\sum a^3 \sin(B - C) = 0.$ (Edit from comment: $a$ is the length of side $BC$. $B$ and $C$ denote angles at vertices $B$ and $C$ respectively.) How can I solve this problem? Any tips ...
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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 : ...
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1answer
24 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 $I_2$ ...
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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
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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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Circumcentre of three points X, Y, Z, given distance from each to points A and B

I'm racking my brain trying to figure out where to start on this, and it's been too many years since working on these kinds of problems. I have six measurements which I'd like to use to calculate a ...
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1answer
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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 ...
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Euclid geometry-triangle inequality [on hold]

How to prove that the sum of the diagonals of a convex pentagon are larger than the scope of the pentagon?(using that a+b>c in triangle?
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1answer
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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 ...
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1answer
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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 ...
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3answers
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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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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 ...
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2answers
55 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 ...
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1answer
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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
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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
38 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 ...
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1answer
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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
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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
221 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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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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+150

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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Special Right-angled Triangles

How do I solve number 11-12? I'm in 2nd year junior high from Indonesia. I don't understand this at all, since I was absent during the first class about this. I do know that there are 2 special right ...
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2answers
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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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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
31 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
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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
38 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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35 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 ...
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1answer
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Which of the following are the correct angle measures for angles 1 and 2 in the triangle shown below?

http://www.explorelearning.com/ELContent/gizmos/ELMath_Deliverable/ExplorationGuides/Geometry/images/EL_GEO_TriSum6.gif A. mangle1 = 43°, mangle2 = 137° B. mangle1 = 137°, mangle2 = 43° C. mangle1 ...
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1answer
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Given a particular triangle that has been constructed, I want to prove that one of the angles must be $> 45$ degrees. [duplicate]

Suppose you are given an acute triangle $XYZ$ with the following properties: At $\angle XZY$, the $\angle$ bisector is drawn and extended all the way to $XY$. Lets call the point where it intersects ...
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2answers
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Two triangles with two equal sides and equal area will have the third size also equal?

Consider two triangles $\triangle abc$ and $\triangle def$ such that $ab=de$ and $ac=df$.Also area of $\triangle abc$ is equal to area of $\triangle def$.Now draw $cm$ perpendicular to $ab$ and $fn$ ...
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Two questions about triangle that blocked at rectangle…

The area of the triangle is equal to the half area of the rectangle? The center point of the triangle is same as the center point of the rectangle? About 2 - if not, how do I calculate the center? ...
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2answers
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Calculating the perimeter of triangle inside of a circle

In triangle $DCB$, $BC = 10$ and is also the diameter. If the area of triangle $DCB = 11$, then determine the perimeter of the triangle. I am a little stuck on this problem. I tried using the sine ...
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2answers
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Triangle in Triangle

I have the lengths of three sides of an acute triangle ABC as shown below. Assume a point P on the side AB such that, if Q is the projection of P onto BC, R is the projection of Q onto CA, P becomes ...
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1answer
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Will two triangles with two equal sides and equal area have same altitude

Consider two triangles ABC and DEF.AB=DE and AC=DF .Also area of triangle ABC is equal to the area of triangle DEF.If we draw an altitude (to one of the equal sides) in both triangles, is it(altitude) ...
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1answer
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What is the measurement of side KL in a scalene triangle

Given: $\triangle FGH\cong \triangle JKL$ What is the measure of side 2 on triangle 2? Here is what I have. Triangle 1: Side 1: Unknown Side 2: Unknown Side 3: 12 Triangle 2: Side 1: Unknown ...
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How to prove this infinite D$_{3\text{h}}$ prism has $0$ volume?

When we build this infinite D$_{3\text{h}}$ prism out of a stack of Nicomachus' triangles and then weight the edges to the differences between the vertices, the prism has $0$ volume. How can we prove ...
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1answer
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Perimeter Of A Simple Triangle

Here in $ \triangle ABC$ $ AC=4 , DE= EF =1, \angle ABC=90^{\circ} $. The perimeter of the triangle $ \triangle ABC$ can be written as $ \sqrt {m } + n $ where $m$ and $n$ are non-negative ...
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Are there some special properties in Pascal's triangle? [closed]

Take a look about it on this website talking about my question. It shows that there are special properties in this triangle, but are those true? The first question I need to ask involves the ...
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1answer
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Finding functions in Inscribed Triangle

If we have a circle of radius $R$ around center $O$ and its inscribed triangle $XYZ$ that is acute as well as scalene. $XY$ is the longest side. $XA,YB, ZC$ are the altitudes of the triangle $XYZ$. ...
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1answer
34 views

Meaning of “circumference”

I am French and I have to solve a math problem written in English. The wording is the following : " In triangle ABC, the angle bisector of angle A intersects line BC at D and the circumference of ...
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1answer
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I need help with this geometry question.

Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to ...
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Triangle Section Side Lengths

Point $D$ is on side $BC$ of $\triangle ABC$, with $AB=3$, $AC=6$, and $\angle CAD = \angle DAB = 60 ^{\circ}$. What is the length of $AD$?
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Side Section Lengths in a Right Triangle

Right triangle ABC has its right angle at C. Let M and N be the midpoints of AC and BC, respectively, with AN=19 and BM=22. What is AB?
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Area of a triangle inside a larger triangle

It's been a while since I've done any geometry so I'm a bit confused by this question. We have a triangle $\triangle PQR$ whose total area is $90 \mathrm{cm}^2$. Another triangle $\triangle PTU$ is ...
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2answers
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Finding Orthocenter in Coordinate Geometry

If a triangle is formed by the equations \begin{gather}2x+3y-1=0\\ ~~x+2y-1=0\\ ax+by-1=0\end{gather} and has its orthocentre at origin, then what are the values of $a$ and $b$? (Please also tell me ...
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Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...