Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
10,045 questions
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Semicircle with tangent and perpendicular: prove that DE·BC = CD·R
Problem
Given a semicircle with diameter AB = 2R and center O. Let C be a point on the
extension of AB beyond B. From C, draw a tangent CD to the semicircle, touching
it at point D. The perpendicular ...
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1
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Explain this seeming contradiction in Euclid Book 1 Proposition 16
This proposition has been concluded without the use of the parallel postulate, because the first time Euclid invokes the parallel postulate is in I.27. Thus, it should apply to all geometries ...
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Draw an isosceles triangle equal in area to a triangle ABC, and having its vertical angle equal to the angle A
I am trying to solve the question
Draw an isosceles triangle equal in area to a triangle ABC, and having its vertical angle equal to the angle A.
I have tried to approach the problem from backwards (...
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5
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Prove that triangle BNC is isosceles in a 30-60-90 construction
Given a right triangle $ABC$ with $\angle A=90°$ and $\angle B=30°$. On the extension of side $CA,ドル we take point $D$ such that $AD=AC/2,ドル and on the interior of side $BC,ドル we take point $E$ such that ...
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3
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Find Angle Without Using Trigonometry
Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle BAC = 108^\circ$. A point $M$ lies inside triangle $ABC$ such that
$$
\angle MAB = 30^\circ \qquad \text{and} \qquad \angle MBA = 12^\circ....
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1
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Minimizing perimeter with fixed area
Problem: Wasim has 6ドル$ poles and a huge rope. He was asked to occupy a
plot of land measuring 96ドル\sqrt{3}$ square meters in the middle of a
very large field. In order to occupy the land, Wasim has to ...
3
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2
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323
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What is the nature of a triangle for which the square of the diameter of its circumcircle is equal to the sum of the squares of two of its sides?
Fifty years ago , when I was in college, our teacher , Mrs Marie -Jo , gave us this homework assignment for the holidays :
" Find the two types of triangles such that the square of the diameter ...
4
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1
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Does the Radical Axis of the Circumcircle and the Miquel Circle of a Cyclic Quadrilateral Coincide with the Steiner Line?
A few months ago, while experimenting with GeoGebra, I came across what looks like an interesting property:
In a cyclic quadrilateral, the radical axis of the circle through the four vertices and the ...
2
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1
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Help Revising a 2-Column Proof for Euclid's Elements I.7
This is my working 2-column proof for Book 1 Proposition 7. I would be remiss in saying that this is completely foolproof. One question is how we are to formulate a proof by contradiction within the ...
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Struggling with geometry for weird reasons. [closed]
I have this book I'm trying to be good at : Stanford problems.
I do well or good enough at problems involving only numbers but I can't do the geometry problems for the life of me. Not even after ...
2
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1
answer
103
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Triangle with equidistant centers.
Consider $\triangle ABC$ and let $H,ドル $I,ドル $O$ be its orthocenter, incenter and circumcenter respectively. Show that:
$$OH \geq OI$$
$$OH \geq HI$$
I stumbled on this properties while experimenting ...
6
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3
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616
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Find a synthetic proof to an old problem .
I found this problem in a French paper translated from Arabic in 1927 by an author named Al Bayrouni . I wonder if it can be found in one of Archimedes' works . Here is the statement :
ABC is a ...
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0
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
...
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1
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Random points in a sphere
I have to randomly put $N$ points in a sphere of radius $R$ in a way that the distance of every point from the other points is greater than or equal to $r_0$. Is there an algorithm to solve this ...
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3
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What is the measure of the angle between the two diagonals of this trapezoid?
The attached figure represents a trapezoid ABCD with four angles indicated.
My objective is to calculate the angle x formed by the two diagonals AC and BD.
Using GeoGebra, I found that x is almost 106°...