Questions tagged [triangles]
For questions about properties and applications of triangles.
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Alternate Proof for Sum of Sides of a Triangle Inequality
I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality.
Note that I am just a high school student and feel free to correct me wherever if I am wrong.
Statement/...
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In the convex quadrilateral $ABCD$ Assuming that $\angle BCD< 90^{\circ}$, prove that:$\angle DAB< 90^{\circ}$
In the convex quadrilateral $ABCD$, with its side lengths $AB$, $BC, CD$, are $25, 39, 52$, and $DA$ $60$ units, respectively. Assuming that $\angle BCD< 90^{\circ}$,
prove that:$\angle DAB< ...
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Proof of Thomson cubic pivotal property without coordinates
The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
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Parallel line equation
I want to incorporate 2 diagonal lines in a logo design. The lines have to be parallel to each other and have to be exactly 0.5 inches apart when measured perpendicular. The upper point of Line 1 has ...
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What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius
What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius 1?
For n = 1, the triangle is equilateral.
For n = 2, we have 2 isosceles right triangles sharing a ...
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Proving Symmedian intersects intersection of tangents
I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians.
Proposition 4.24 says:
Let $X$ be the intersection of the tangents to ...
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What is the minimum value of $a+b-c$ in a triangle with a fixed area?
Let $\Delta$ be the fixed area of a triangle inscribed inside on a fixed circle of radius $R$. The sides of the triangle $(a,b,c)$ are unknown. We want to estimate a the lower bound of the triangle ...
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acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$, then $ABC$ is right-angled. [duplicate]
Given that the acute angles $\alpha$ and $\beta$ of the triangle $ABC$ satisfy the condition $\sin^2 \alpha + \sin^2 \beta = \sin (\alpha + \beta)$. Prove that the triangle $ABC$ is right-angled.
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Calculate the length of segment $AD$.
Given a triangle $ABC$ and its circumscribed circle, point $E \in BC$. Let $D$ be the intersection of the circle and line $AE$ (see the figure). Also, let $|AB| = |AC| = 12$ and $|AE| = 8$. Calculate ...
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Finding all empty triangles of a plane
I have a set of $N$ points ${(x_i,y_i)}_{i=1,...,N}$. I am looking for an efficient algorithm to find the set of all empty triangles (i.e., that do not contain any points).
The brute-force method that ...
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Find the total area of two triangles within a square
I solved this but others have conflicting answers, I'd love some validation:
Total area of the green triangles.
Please show your work on how to solve this - lots of Pythagoras and more needed.
Ok, ...
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calculus optimization problem: rectangle inscribed in a triangle.
I have a solution to the problem below from my course materials, but I cannot understand where I went wrong with my own attempt at a solution. Any advice much appreciated.
Problem:
Given a right ...
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Find the segment BT in the triangle inscribed below
In the figure, $AB.BC = 60$ and $BT.TP = 40$.
Calculate BT with B and T tangency points.
(Answer:$2\sqrt5$)
I try:
$AT.TC = BT.TP \implies AT.TC = 40$
$AM.AB = AT.AC$
$AT^2 = AM.AB \implies AT^2 = AT....
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Isosceles triangle calculations
We have that if $𝑆 ⊂ ℝ^2$ is a set of $𝑛$ points in the plane, with no three points on a common line, then there exists a point $𝑎 ∈ 𝑆$ that determines at least $(𝑛 − 1)/3$ distinct distances to ...
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Understanding the geometry behind finding the area of a triangle defined by three vertices in three space
I am self-studying linear algebra using Jim Hefferon's Linear Algebra textbook (published in 2020). As I was making my way through the determinants section, I stumbled upon the following question:
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