Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
10,184 questions
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For three circles, if three paired common tangents concur oddly outside their contact segments, prove the other paired tangents concur too.
For three circles, draw a pair of common tangents for each pair of circles. If the three common tangents in one set are concurrent, and if their point of concurrence lies outside the contact-point ...
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A question about a pair of triangles in a triangle
An example of this phenomenon.
Given triangle $\triangle{ABC}$, define another triangle $\triangle{A'B'C'}$ such that A' lies on $\overline{BC}$, B' lies on $\overline{AC}$, and C' lies on $\overline{...
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How to prove that four points are concyclic under these given conditions?
I've stumbled upon the following problem that made me curious for quite a while. Below is one of many possible diagrams of the problem in question.
Given that $ABCD$ is a parallelogram with acute ...
- 429
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Existence of this trick-of-the-eye figure in 3-dimensional space
Prove or disprove: There exist eight points $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ in $\mathbb R^3$ such that $\square ABCD$, $\square BCGF$, $\square EFGH$ and $\square ADEH$ are convex ...
- 1,860
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Is the limiting incenter of a degenerating tetrahedron the Radon point of the four limiting coplanar vertices?
Let $T_n$ be a sequence of nondegenerate tetrahedra in $\mathbb R^3$, with labeled vertices
$$
A_n,B_n,C_n,D_n.
$$
Assume that
$$
A_n\to A,\qquad B_n\to B,\qquad C_n\to C,\qquad D_n\to D,
$$
where $A,...
- 551
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Elementary geometry: showing that four points are coplanar
Suppose we are given four coplanar points $A$, $B$, $C$, $D$, in three-dimesional Euclidean space.
Suppose that we are also given four points $A'$, $B'$, $C'$, $D'$, such that $\left| AB \right| = \...
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How to map square roots as a linear progression on a circle?
I have been exploring geometric constructions with roots. It seems that the fact that the circle is the locus of precise square root dispositions is often overlooked, as I haven't seen it used much in ...
- 483
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Concyclicity of Fermat points in a cyclic quadrilateral
About six months ago I came up with a nice property related to Ferma points and circular quadrilaterals, but I couldn't prove it:
Let $ABCD$ be a cyclic quadrilateral. For each vertex, consider the ...
- 4,843
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Non-concyclicity of the circumcenters of complementary triangles in a quadrilateral
About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, ...
- 4,843
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How to construct the following isosceles trapezoid with compass and a straightedge?
Given:
$ABCD$ is an isosceles trapezoid with $BC \parallel AD$.
$MN$ is a segment such that $M\in AB$, $N\in CD$, $BC\parallel MN\parallel AD$.
$AM : MB = DN : NC = 1 : 2$.
$MN = AB = CD$.
$O$ is a ...
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How can you demonstrate that $x = a + b + c$ without using trigonometry?
Show that $x=a+b+c$
I solved this problem using trigonometry, however, I want to solve it using only geometric constructions.
I tried to create a triangle with sides $x$ and $a+b+c$ and show that the ...
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Calculating the radius of a semicircle inscribed in a triangle
I've just rediscovered an old problem I first encountered over half a century ago: It's the problem number $191$ from Julius Petersen's famous work, whose initial statement is as follows:
In a given $...
- 3,518
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On Line-Plane Perpendicularity in Euclidean Geometry
Due to a course of euclidean geometry that I enrolled to complete my graduation degree, in which we study plane euclidean geometry from the axiomatic point of view, I've decided — for fun! — to try on ...
- 1,298
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Determining a quadrilateral up to similarity with four angles
A triangle, up to similarity, is completely determined by two of its internal angles.
A quadrilateral can be divided into two triangles, and those two triangles are independent from each other; thus, ...
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Coaxality of three circles in a complete cyclic quadrilateral
Let $ABCD$ be a cyclic quadrilateral. Consider the three intersection points of its pairs of opposite sides:
$P = AB \cap CD$,
$Q = AC \cap BD$,
$R = AD \cap BC$.
These three points form the diagonal ...
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