For questions about properties and applications of triangles

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Angles sum in a triangle on the x- axis.

$\angle BCA=90$ degrees I probably do not understand the concept of angle sum in a triangle but here is the thing. $\angle BAC$ is negative by convention. So is $BCA$ going to be greater than 180 ...
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Congruence of triangles: SSA criteria

It is well known that this criteria does not work in general. I am trying to answer to the following question if two triangles have two sides and the angle NOT between them equal, they are either ...
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1answer
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In a triangle $\Delta ABC$, let $X,Y$ be the foot of perpendiculars drawn from $A$ to the internal angle bisectors of $B$ and $C$

In a triangle $\Delta ABC$, let $X,Y$ be the foot of perpendiculars drawn from $A$ to the internal angle bisectors of $B$ and $C$. Prove that $XY$ is parallel to $BC$. It works for an equilateral ...
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Mid-sections and angles

In the triangle $ ABC $, whose $ AC> BC> AB $, on the sides $ BC $ and $ AC $ chose the point $ D $ and $ K $, respectively, so that $ CD = AB $, $ AK = BC$. Points $ F $ and $ L - $ midpoints $ ...
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Help required to prove a question on triangles, collinearity and cyclic quadrilaterls

In an acute $\triangle ABC$ $D,E,F$ are feet of perpendiculars from $A,B,C$ respectively. The perpendiculars from $F$ to $BC,AD,CA,BE$ intersect them at $X,Y,Z,V$. How do I prove that $X,Y,Z,V$ are ...
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Geometric Construction : Construct a Triangle given 3 heights .. [closed]

Given 3 heights : $h_1=5\mathrm{cm}$ ; $h_2=7\mathrm{cm}$ ; $h_3=8\mathrm{cm}$ ... It is required to draw that triangle using only compass and ruler ! N.B.: It is not allowed to calculate the area ...
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Area of a triangle from vector coordinates of vertices in 3D

I have three vectors $v_1, v_2$, and $v_3$, giving the vertices of a triangle. The $z$-coordinates are the same, so the $(x,y)$-coordinates alone give the vertices of an identical triangle in the ...
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1answer
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Reciprocal Altitude Theorem

If $ABC$ is a triangle such that $D \in (BC) $ , $AD^2 =BD \cdot CD$ and $AD= \frac{AB \cdot AC}{BC} $ show that $ABC$ is a right triangle. I tried to solve it with Stewart Theorem but the calculus ...
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2answers
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What does “radius” mean when talking about reference triangles?

I'm watching this trig tutorial and at several points the guy refers to the hypotenuse of the triangle as the "radius" and explicitly writes $2 = r$. To be clear, it's a $30^\circ - 90^\circ - ...
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1answer
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Orthocentre of a triangle [closed]

I just want to know what the orthocentre of a triangle is? How do you define it? It appears in a past paper so i just need the definition. Thanks!
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Triangles with no common side in a polygon

There are n sides of a polygon(where $n>5$). Triangles are formed by joining the vertices of the polygon. How many triangles can be constructed with no side common to the polygon? My try: Total ...
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1answer
43 views

Determine if a point is inside a subtriangle by its barycentric coordinates

See this figure ABC is a triangle. R is a point inside that triangle, specified by its barycentric coordinates. w is a scalar. We mark the points B' and C' such that BB' == w and AB' == AB - w ...
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1answer
39 views

Comparison of triangle areas

Let $ABC$ be a triangle in the plane and $X,Y$ and $Z$ points on the segments $BC,CA$ and $AB$. $X,Y$ and $Z$ are not identical to any corner of $ABC$. Additionally, for a given $X$, let $P$ be the ...
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2answers
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Ratio of lines intersecting in a triangle

In $ABC$ triangle $M$ is the mid point of $BC$ and $N$ is a point on $AB$ that such that $AN:NB = 2:1$. $AM$ and $CN$ are intersected at the point $D$. What is the ratio of $AD:DM$?
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Why is not possible to draw this triangle?

Why is it not possible to draw triangle $DEF$ with $EF=5.5cm$,$\angle E=75^0$ and $DE-DF=1.5cm$?(I used this method for ...
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1answer
25 views

3 coordinate of a right triangle

My math and geometry skills are extremely rusty at this point in my life. Could someone walk me through a formula to find the 3 coordiate point's (XY) of this right triangle? I know the distance of ...
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1answer
53 views

Why aren't area of triangle not same when calculated by different methods in this case

I came across a question today. Two mutually perpendicular straight lines through the origin forms an isosceles triangle with the line $2x + y = 5$. Then the area of the triangle is ? I know ...
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1answer
167 views

Construct a triangle with its orthocenter and circumcenter on its incircle.

Construct $\triangle ABC$ such that its orthocenter ($H$) and circumcenter ($O$) are on its incircle. I've tried something by inverting everything WRT circumcircle but don't have proper idea... ...
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1answer
109 views

What is the “dropoff” to the ground from the eye line of an observer straight across a curved globe earth?

Please note that we assume the observer's eye line is exactly at sea level (0 inches) and we are assuming a perfect spherical earth with no atmospheric effects. The idea here is an alternative ...
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Univalent triangle inequality [duplicate]

$|Z_1| = | \frac{v(1+\alpha) + \sqrt{v^2(1+\alpha)^2-4\alpha}}{2}|$ Triangle inequality |x+y|=|x|+|y| Where x= $\frac{v(1+\alpha)}{2}$ and $y= \frac{\sqrt{v^2(1+\alpha)^2-4\alpha}}{2}$ I've been ...
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1answer
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Triangle inequality univalent

$|Z_1| = | \frac{v(1+\alpha)+ \sqrt{v^2(1+\alpha)^2-4\alpha}}{2}|$ I know that using triangle inequality method $|Z_1|$ is: $|Z_1|= |\frac{v(1+\alpha)}{2}| + ...
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1answer
44 views

Rational distances in triangle

Given triangle with sides of length $3, 4$ and $5$ prove that if $P$ is a point inside the triangle, then rationality of sum of distances from the point $P$ to the vertices implies rationality of sum ...
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1answer
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Equation of circumcircle formed by $x^2+y^2+2gx+2fy=0$ and $2x+y=1$

​​​The equation of circumcircle of triangle formed by lines $7x^2+8xy-y^2=0$ and $2x+y=1$ is $x^2+y^2+2gx+2fy=0$ ,then find $g$ and $f$ I thought if I make equation of circle homogeneous with the ...
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2answers
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If $a^2+b^2=5c^2$ where $a,b,c$ are the sides of a triangle, prove that the area of triangle is $c^2\tan C$

If $a^2+b^2=5c^2$ where $a,b,c$ are the sides of a triangle, prove that the area of triangle is $c^2tan C$ Let median through $C$ be $CF$. $AF=FB=\frac{c}{2}$ ...
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45 views

Maximize area of a triangle

Given a triangle $\Delta ABC$ in the plane and points $X,Y$ on line segments $BC$ and $CA$, respectively, so that the lines $AB$ and $XY$ are parallel, find the locations of $X$ and $Y$ so that the ...
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1answer
42 views

Maximize distance to closest vertex inside triangle

Question: Let $\Delta ABC$ be a triangle. For any point $P$ inside or on the boundary of triangle, define $d(P)=\min\{\overline{PA},\overline{PB},\overline{PC}\}$. Find the maximum of $d(P)$ (in ...
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1answer
37 views

Does the inner pentagon inside a Robbins pentagon $also$ have a rational area?

The Heron triangle has integer sides and area. The Robbins pentagon is just the generalization: it also has integer sides and area. The example below has sides $78, 126, 66, 50, 32$ and area $A_R = ...
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1answer
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Trigonometry: Find points coordinates in equally arms triangle

h have a equally arms triangle. The angle on point C is not 90 degrees. I have: The coordinates of point $C(C_x, C_y)$ The coordinates of the end point of $h$, $H(H_x, H_y)$ The length of $C$ ...
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1answer
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Plane geometry problem, Suppose ABP,BCP,CAP have same area&perimeter…

I'm trying to solve following geometry question, but it is quite challenging.(at least for me!) Thanks for your help in advance. On plane, there is some triangle ABC. Also, there is a point P ...
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Minimize area of a triangle

Let $\Delta \mathrm\,{ABC}$ be a triangle in the plane and $X,\, Y,\,Z$ be points on sides $BC,\, CA,\,AB$, respectively. If lines $XY$ and $AB$ are not parallel, there is a location for $Z$ ...
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1answer
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Prove that if the altitude and median of a triangle form equal angles with sides then the triangle is right.

Problem statement: Prove that if the altitude and median drawn from the same vertex of a nonisosceles triangle lie inside the triangle and form equal angles with its sides, then this is a right ...
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How do you find the base of an isosceles triangle when given the legs only?

This is an SAT math problem, and it's really confusing me. http://i.imgur.com/VuIGVdu.png I am completely lost. Do I have to use trigonometry? I know that the angles of the triangle are 30, 30, and ...
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Prove that the sum of angles is equal to 90° using complex numbers

On the picture, we see three squares: $ABGH$, $BCFG$ and $CDEF$. Prove that the sum of angles: $\angle DAE$, $\angle CAF$ and $\angle BAG$ is equal to $90°$. The real problem is that we have to ...
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1answer
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Find $\angle B$ if $AD=\frac{abc}{b^2-c^2}$

If AD is median and $AD=\frac{abc}{b^2-c^2}$ $[b>c]$ and $\angle C=23^{\circ} $. Find $\angle B$ Is this information sufficient to find $\angle B$? I tried using sine rule in triangle $ADC$ and ...
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1answer
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prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
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Prove that the area of a triangle DEF is correct.

There's any triangle ABC. First player 1 has to set D on AB so that in the end the triangle DEF has the highest possible area. Second player 2 has to set E on BC so that in the end the triangle has ...
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Triangles - sin, cos etc. [closed]

I know this is a quite simple question for most of you out there. However it has been a little troubling for me, and would like to get a little help if possible. I have a triangle $ABC$ where I know ...
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A curious triangle inequality

Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that \begin{equation} |PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|. \end{equation}
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Find the sides of a right triangle formed by connecting two other right triangles from the center of their hypotenuse.

I have the following sketch of the problem: I need to find the values of $x$ and $y$ in the previous drawing. The hypotenuses of both black triangles are of equal length and the red triangle is a ...
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2answers
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What trigonometric identity makes the method of triangulation work?

I've read the article on Wikipedia, but I don't get how to construct the relationships between sides and angles to reach a solution for the distance between two points. All the other sites I read ...
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Proof of Cauchy-Schwarz Inequality 1

In my lecture notes I've written the proof of Cauchy-Schwarz inequality as: Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + ...
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A Quadrilateral and A Triangle in a Trapizium

In the above diagram, $ABCD$ is a Trapizium with $AD || BC$ and $BC \perp AB$ $AB = 20, \; AD = 6,\; BC = 30$ $M$ is a point on $DC$ such that $[ADMB] = [BMC]$, where $[x]$ denotes the area of $x$. ...
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Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
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0answers
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Using oblique projection can you always rotate a triangle to look like an equilateral triangle? [duplicate]

Starting with any triangle using oblique projection, can you view any shape triangle from an angle to see it as an equilateral triangle?
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How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
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3answers
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New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
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1answer
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Find $|CM|$, if $|CA|=a$ and $|CB|=b$. [closed]

Let $O$ be a center of a circle, circumscribed over $\triangle ABC$. Perpendicular, drown from the point $A$ on the line $CO$, cross the line $CB$ in the point $M$. Find $|CM|$, if $|CA|=a$ and ...
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1answer
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Prove that $MN = \dfrac{|b − c|}{2}$

In triangle $ABC$, point $M$ is the midpoint of $BC$ and $N$ is on the angle bisector of $\angle A$ such that $MN \parallel AB$. Prove that $MN = \dfrac{|b − c|}{2}$. Attempt: I drew it out and ...
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2answers
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Geometry/ Triangles problem

I have been struggling with this problem, and I think it should be possible to solve but right now I cannot find how. Given two coordinates/points (x1,y1) and (x2,y2) The angle d1 with the ...