Questions tagged [triangles]
For questions about properties and applications of triangles.
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Proof using Converse of Thales Theorem for isosceles right-angled triangle
Let $ABC$ an isosceles right-angled triangle with the right angle at $C$. Suppose that the points $D$ and $E$ lie outside the triangle on the half-line $AC$ and $CB$, respectively (see picture).
Let ...
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Maximum area a traingle can have which can fit inside a circle of radius $r$?
So what is the maximum area of a triangle which can fit inside a circle of radius r?
My first approach: We know that $\text{ Circumradius }=\frac{abc}{4×\text [area-of- triangle}$ (here abc are side ...
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Determination of the Area of an Internal Triangle in a Scalene Triangle with Division Points and Their Intersection [closed]
In 2 hours I couldn't solve the problem help pls i need the answer
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2-D scalene obtuse triangle trigonometry.
I am struggling with this trigonometry question:
question
I tried using the cosine law with angle DBC
$a^2 = b^2 + c^2 - 2bc \cos A$
but you need to know the measure of the angle.
In terms of the ...
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Largest Area Triangle in the Vesica Piscis
I can place any three points in or on a vesica piscis1. I wish to find the triangle of maximum area. I know the area of the vesica piscis is $(\frac{2π}{3}-\frac{\sqrt{3}}{2})d^2$ (where d is the ...
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Knowing a side, the inradius, and the circumradius of a triangle, find the other two sides [closed]
I need help with this easy triangle problem:
We know:
One of the sides a = 16 cm.
The inradius r = 6cm.
And the circumradius R = 17 cm.
That's all.
We must find the lengths of the other two sides.
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Determine the angle $\angle DEC$ in a triangle (Euclidean Geometry)
Any ideas how to find the angle $\angle DEC$ in the following situation shown in the image:
In the above figure we have that $\angle BAC = 90, \angle ABD = \alpha, \angle DBC = 2\alpha$, and $\angle ...
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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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