# Tagged Questions

For questions about properties and applications of triangles

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An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
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### 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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### Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
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### When does there exist a point with a given ratio of distances to the vertices of a triangle?

I have the triangle ABC and an unknown point P not necessarily inside the triangle. Also, I have three lengths (...
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### “Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
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### Area of $A'B'C'$ is to area of $ABC$ is $\frac{(m-n)^2}{m^2+mn+n^2}$

In the sides $BC,CA,AB$ are taken three points $A',B',C'$ such that $BA':A'C=CB':B'A=AC':C'B=m:n$.Prove that if $AA',BB',CC'$ are joined they will form by their intersections a triangle whose area is ...
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### Proof of equilateral triangle given angles

Let's say we start with a scalene triangle ABC, with no given angle measures or side lengths: Then, we add 3 Isosceles triangles adjacent to this one, given that they have angle measures 30*30*120,...
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### Number of triangles created after $n$ folds of a square

My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard. Fold a ...
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### How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
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### Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a triangle. ...
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### Prove that the maximum volume of a triangular-base prism is $\sqrt{\dfrac{K^3}{54}}$ where K is the area of three triangles containing a vertex A

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and ...
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### Find the lengths of sides of a right triangle

If I know the length of the hypotenuse of a right triangle ONLY, is it possible to find the lengths of the remaining 2 sides of that same triangle?
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### Find the length of the side of a right angle triangle inside a circle

Hello Stack Exchange. I have a question which has really been preventing me from making a certain program.In my program I need to find the length of AC using only AB and BD.The triangle is right-...
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### Trigonometric roots of a cubic

Let the product of the sines of the angles of the triangle is $\frac{2}{3}$ and the product of their cosines is $\frac{1}{9}.$ If $\tan A$ , $\tan B$ and $\tan C$ are the roots of the cubic, find the ...
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### Condition for the existence of a triangle

Could you please explain and solve this problem for me? I would really appreciate it. The more depth of explanation, the better. Let $a$, $b$, $c$ be non-collinear vectors. Show that the necessary ...
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### split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
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### Trigonometric question involving similar triangles

The $\alpha$ angle is supposed to be 90.00 degrees. Couldn't get it right with the graph tool! I'm trying to understand why the CK segment isn't $$CK = CJ*cos(Y) \\ CK = CJ*\frac{CA}{CB}$$ and ...
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### Area of triangle constructed from the medians of other triangle

We have triangle ABC of area P. Is it possible to compute the area of a triangle with sides "made" from medians of the triangle ABC in terms of P I'm looking for some hints maybe,
I have found the following problem: There is an acute $\triangle ABC$. Denote its circumcircle as $\omega$. The angle bisector of $\angle BAC$ intersects $BC$ and $\omega$ in points respectively $A_1$ ...