# Tagged Questions

For questions about properties and applications of triangles

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### 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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### 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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### 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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### On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $$|PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|.$$
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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 ...