For questions about properties and applications of triangles

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Geometry: Determining the length of a side of a triangle [closed]

Triangle $ABC$ has all sides of integral length. Vertex $A$ is at $(0,0)$, $B$ lies on the line joining $(0,0)$ and $(3,6)$ and $C$ lies on the line joining $(0,0)$ and $(2,-1)$. Two of the three ...
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1answer
32 views

Finding the length of a side of an equilateral triangle

There is a large right isosceles triangle with a hypotenuse length of $24$. Inside the triangle is an equilateral triangle with a vertex on the midpoint of the hypotenuse. If the length of each side ...
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1answer
31 views

Solving for length of an unknown side of a triangle.

I have been given the figure below: Figure (click me). I know that $AD=20-x$ and $m\angle ACD=m\angle BCD$. How can I set up a ratio also knowing that $AC=11$ and $BC=14$ in order to find $x$? ...
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3answers
40 views

Find the measure of a side and an angle.

In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$. Given that $AB=BC$, find $AD$ and $m\angle ABC$. Here is the figure: I am inclined to say that since $\overline{AB}\simeq ...
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1answer
37 views

How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image: I know coordinates of an example P point. Say, they are equal to (8,8). I also ...
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4answers
63 views

Properties of Equilateral Triangles in Circles

If there is an equilateral triangle in a circle, would the midpoint of any of the 3 sides be half the radius? e.g if the radius was 6 and at the midpoint of the triangle (call it B) would center to ...
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1answer
26 views

Proof formula for triangle: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$

given is a triangle with corners A,B,C and corresponding sides a, b, c. D is a point somewhere between A and B. I have to proof: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$ Unfortunately I have no ...
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1answer
27 views

Find the width/height of a triangle given a side length and two lines

I'm a programmer and I came across a math problem in my current project that I can't figure out. My situation looks like this. Everything in black is known or I know how to figure out. ...
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0answers
108 views

Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
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1answer
117 views

Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the ...
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0answers
19 views

Division of Solid Angle When Subdividing Spherical Triangle

Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge: ...
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1answer
39 views

Proving the Pythagorean Theorem with just variables

I basically have just three problems: a) How many similar triangles can you find in the figure below? b) Use part a) to prove the Pythagorean Theorem (note: to prove something you do need to ...
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4answers
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to ...
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1answer
36 views

Area of Traingle Problem

In Triangle $DEF$, $P$ is mid point of $EF$ and $Q$ is the midpoint of $DP$. The area of triangle $DQF$ is $6 \ cm^2$. We need to find the area of triangle $EQF$. I tried many ways to solve it but ...
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3answers
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Triangle: Finding $x$ and $y$ (2 sides are given) - 6th grade

I am helping my son with this homework and I was wondering if I can get a tip or few. The question is: Write and solve equations to determine the values of $x$ and $y$. (see picture attached) The ...
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2answers
40 views

value of X and Y from triangle

my son is in 6th grade and i am trying to help him solve this problem. but i want to understand so i can teach him. Write and solve equations to determine the value of x and y . triangle is given ...
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1answer
28 views

For a given triangle, prove that $DL=DM$

In a triangle $ABC$, $D$ is midpoint of the side $BC$. Through the point $A$, $PQ$ is any straight line. The perpendiculars from the points $B$, $C$ and $D$ on $PQ$ are $BL$, $CM$ and $DN$ ...
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1answer
54 views

Finding max perimeter of triangle of three circulating points

I'm thinking a plane geometry problem, and it seems quite puzzling. Here it is. Question: Consider three concentric circles with radius 3, 5 and 7 each. and construct a triangle by picking one ...
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0answers
25 views

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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1answer
108 views
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30 views

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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1answer
36 views

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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1answer
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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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3answers
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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
38 views

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
17 views

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
20 views

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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3answers
35 views

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
39 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
38 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
19 views

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
46 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
165 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
107 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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0answers
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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
40 views

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
26 views

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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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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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
41 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
36 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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70 views

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
29 views

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 ...