Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
10,051 questions
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Divide a right triangle into three quadrilaterals of equal area .
The goal is to find the point $M$ inside a given right triangle $ABC$ such that $\operatorname{Area}(APMN)=\operatorname{Area}(CQMP)=\operatorname{Area}(BQMN)$, where $N$, $P$, and $Q$ are the ...
- 1,898
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Collinearity of $M', H, N'$ where $M', N'$ are reflections of midpoints across sides of the intouch triangle
Problem Statement:
Let $ABC$ be a triangle. Let $AH$ be the internal angle bisector of $\angle BAC$ (with $H \in BC$). Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Let ...
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Perpendicular from vertex in square - angle problem
Let ABCD be a square and Z the midpoint of BC. From vertex A, draw a perpendicular to line DZ with foot E. Draw segment EC. Find ∠ZEC.
Context:
I was studying parallelograms and their properties, ...
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Using geometric constructions to solve algebraic problems (in Euclid and Descartes)
Long-time listener, first-time caller here. I've got what I guess is a history of math question. I've been teaching a college course on Euclid's Elements (surprisingly fun, btw). I've been using David ...
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Relation between cuts of ellipsoid by lines and the ellipsoid itself
I stumbled across a problem regarding ellipsoids and got stuck trying to prove this relation between ellipsoids and their sections. Consider an ellipsoid $E$ written as
$$
E = \left\{x \in \mathbb{R}^...
- 1,870
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Find $\frac{EF}{EG}$. A contest geometry from Bangladesh
Problem:
In the figure, $ABCD$ is a square. A circle drawn through the vertices $C$ and $D$ and the middle point $E$ of $AB$. Assume the circle intersects $AD$ at $F$. And $G$ is a point on the ...
- 789
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Dissecting a "Line-Perpendicular Triangle" (side ratio 1:2) and finding the distance relationship.
I came across a geometry problem from a Chinese Grade 9 math exam that involves a specific type of triangle defined as a "Line-Perpendicular Triangle" (线垂三角形). I am stuck on the construction ...
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How to prove that the locus of point $Q$ is a circle?
I've encountered the following problem:
Segment $AB$ is a diameter of the black circle $M$. The blue circle $T$ passes through point $B$, and $P$ is a point on the blue circle $T$. Draw line $BP$, ...
- 341
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4
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Is there something missing with this simple geometry problem?
I'm currently doing an exercise form my geometry book. The question is asking for the volume of the pyramid $N.ABCD$ (i.e. a pyramid of base $ABCD$ and with the tip $N$). The construction is as ...
- 695
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Semicircle with tangent and perpendicular: prove that DE·BC = CD·R [closed]
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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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$, we take point $D$ such that $AD=AC/2$, and on the interior of side $BC$, we take point $E$ such that ...
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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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Minimizing perimeter with fixed area
Problem: Wasim has $6$ poles and a huge rope. He was asked to occupy a
plot of land measuring $96\sqrt{3}$ square meters in the middle of a
very large field. In order to occupy the land, Wasim has to ...
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