# Tagged Questions

For questions about properties and applications of triangles

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### At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
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### When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
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### How many triangles can be created from a grid of certain dimensions?

How would you determine how many non-degenerate triangles can be drawn by connecting points in a $5 \times 5$ grid?
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### Related rates: Find dA/dt of triangle, given d(theta)/dt — Can't come to textbook answer

The question: ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the ...
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### Linear algebra - find all possible positions of the third corner?

An equilateral triangle lies in the plane $x + y - z = 1$ and corners in points $(1, 1, 1)$ and $(2, 1, 2)$. Determine all possible positions of the third corner?
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### Area of convex n-gon using triangles

Suppose we have a convex $n$-gon and a point inside the $n$-gon or on the sides of the $n$-gon, and suppose one extended lines from all the vertices of the $n$-gon to make $n$ triangles with two of ...
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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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### Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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### Prove that $\angle FGH = \angle GDJ$

Let $FGH$ be a triangle with circumcircle $A$ and incircle $B$, the latter with touchpoint $J$ in side $GH$. Let $C$ be a circle tangent to sides $FG$ and $FH$ and to $A$, and let $D$ be the point ...
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### A triangle determinant that is always zero

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
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### Proving Stewart's theorem without trig

Stewart's theorem states that in the triangle shown below, $$b^2 m + c^2 n = a (d^2 + mn).$$ Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
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### How many triangles are there?

The question is how many triangles are there in the following picture? I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I don'...
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### Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides a, b, c. The "square" which I will refer to, from now on, has all its four vertices on the sides of the ...
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### Right triangle on an ellipse, find the area

Beginning note: Please wait until the animations load. The loading might take some time depending on your internet connection. Secondly, the title and the content of the question might not be well ...
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### Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
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### Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
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### How to find surface normal of a triangle

If I have a triangle with $3$ points $P_1, P_2,$ and $P_3$, each with $x, y,$ and $z$ coordinates, how do I find the surface normal $N$ in $x, y,$ and $z$ such that $$N_x+N_y+N_z = 1$$ I'm looking ...
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### The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?
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### probablity of random pick up three points inside a regular triangle which form a triangle and contain the center

what is the probablity of random pick up three points inside a regular triangle which form a triangle and contain the center of the regualr triangle the three points are randomly picked within the ...
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### how to calculate area of 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
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### Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
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### Triangle inequality for subtraction?

Is the following inequality(that looks like the triangle inequality) valid: $|a - b| \leq |a| - |b|$ Why?
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### Prove that the centre of the nine-point circle lies on the midpoint of the Euler line

In $\Delta ABC$, $AD, BE, CF$ are the altitudes and $\Delta A'B'C'$ is the medial triangle. $K, L, M$ are the midpoints of $AH, CH, BH$. Consider the nine-point circle with centre $G$ (not to be ...
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### Find the sides of the triangle.

The triangle with sides $8-15-13$ has a $60^{\circ}$ angle. The triangle with sides $11-35-31$ also has a $60^{\circ}$ angle. Find a triangle $x-y-403$ where $x$ and $y$ are relatively prime positive ...
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