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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

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is it always true that randomly given three line segments of equal length always forms an equilateral triangle.

Is it possible to conclude, if three line segments are equal in length then they always form an equilateral triangle at their common intersection points?
nimmy
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Why do triangles with congruent angles have the same corresponding side ratios?

When I learned this fact, everyone just took it for granted, and no one attempted to prove it. I've seen the proof using the law of sines, but is there a way of proving this without trigonometry?
pavle
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Area of a triangle bounded by diagonal of a square and a second intersecting line

Given the following image, Determine the area of FEC given that the total area is 1 area unit. The correct answer should be 1/12 a.u. but I cannot get all the way to that conclusion. Note, one's not allowed to use sin or cos, which would make for…
Raoul
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A coordinate proof that every polygon with a circumscribed circle is convex

Suppose that we have a polygon $P$ whose vertices are $z_1, z_2,..., z_n$ inscribed in a circle $C$ in the Cartesian coordinate system. Furthermore, suppose that along the perimeter of the circle, $z_1$ is adjacent to $z_2$, $z_2$ is adjacent to…
user555729
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Inscribing two triangles in a parallelogram with specific requirements on areas of triangular regions

$\mathit{ABCD}$ is a parallelogram that is not a rectangle. $E$ is a point on $\overline{\mathit{BC}}$, and $F$ is a point on $\overline{\mathit{CD}}$. $\triangle\mathit{ADE}$ and $\triangle\mathit{ABF}$ are inscribed in the parallelogram, and they…
Adelyn
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To prove that triangles of side $a^n, b^n, c^n$ are isosceles.

Let $a\ge b \ge c\gt 0$ be real numbers such that for all $n \in \mathbb N$ there exist triangles of side $a^n, b^n, c^n$. Prove that the triangles are isosceles. I tried proving it by writing $c^n + b^n \gt a^n$ and when I assumed some values for…
Priyanka
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Question on isosceles triangle

Let $ABC$ be a triangle such that there exists $N \in [AM]$ such that the angles $BAN$ and $CAN$ have the same measure, where $M$ is the midpoint of the segment $BC$. Then $ABC$ is isosceles. One hint?
Almath
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Euclidean Geometry proof

In triangle $ABC$, $AB = AC$ and $A-D-B$ such that $DC = BC$. Prove: $$BC =\sqrt{(AB*BD)}$$. I've tried to use altitudes and the Pythagorean theorem but I'm completely lost on how to solve this problem.
AmandaM
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Area of triangle Problem where two triangles overlap

Let AC and CE be perpendicular line segments,each of length 18.Suppose B and D are the mid points of AC and CE respectively. If F is the intersection of EB and AD ,then the area of triangle DEF is My attempt is Area of $\Delta \space DEF=($ Area…
Saradamani
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About Euclidean geometry

If I have system of axioms from which I can deduce all theorems of absolute geometry, and among this axioms I have some of the variants of fifth Euclid postulate, does it mean that this system will produce whole Euclidean geometry ?
Юрій Ярош
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Angles of intersecting tangent lines of a circle

It's best that I use a diagram to illustrate the question: Is there any interesting relationship, one that can be expressed mathematically, between $\angle CDB$ and $\angle CAB$
John Glenn
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Euclidean proposition 8 of Book I

Euclidean proposition 8 of Book I I'm reading about the Euclidean Elements. What does this proposition mean?
WinstonCherf
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geometry with $T$

Consider an isosceles $ABC$ triangle ($AB=AC$), such that if $D$ is the midpoint of base $BC$, then $BC=AD$. Let $E$ be a point on segment $AB$, such that $DE\perp AB$. Furthermore let $F$ be the midpoint of segment $DE$. Prove that lines $AF, CE$…
Pet123
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Inverse of triangle inequality

If triangle inequality holds for some three sides, does it mean that there exist a triangle with this three sides ? If it is, what is the proof ?
Юрій Ярош
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Axiom in Neutral Geometry

Let $A,B$ be two points on opposite sides of a line $l$. Then the line segment $AB$ intersects $l$. My question is: Using only the 4 postulates of Euclid, is there a way to make precise the meaning of ``opposite sides"? Is the intersection…
Tongou Yang
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