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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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Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex

Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\neq C$. Prove that $AE$ bisects segment $BD$. This is my…
Max
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To prove two angles are equal when some angles are supplementary in a parallelogram

The point 'P' is situated inside the parallelogram ABCD such that the angles APB and CPD are supplementary.Prove that angles PBC and PDC are equal
lokesh sangabattula
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Solve a convex quadrilateral with four sides and equality of two adjacent angles analytically?

Given the length of four sides of a convex quadrilateral and knowing that two adjacent angles are equal, the quadrilateral is determined. I want to know whether there's a formula representing the measure of angles with length of sides. I've tried…
Bei
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How to draw an $405^\circ$ angle?

In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection between the formal definition and the drawing?…
Student
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Relating the incenters of the original and medial triangles.

Let I be the incenter of △ABC. If I is also the incenter of the medial triangle of △ABC, show that △ABC must be equilateral. I'm thinking a place to start would be to show the distance between AC and WU is the same as the distance between AB and VU,…
user181928
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calculate points coordinates on plane from their distances matrix

Given a list of points on a plane is simple to generate a distances list between each pair of points. Pseudo Code: distances = [] for each point_a in points at index_a for each point_b in points at index_b having index_b > index_a dist =…
aleclofabbro
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Convexity of circle in neutral geometry

I am trying to prove that a circle is convex in neutral geometry. i.e. If $A$ and $B$ are inside a circle $C$, than any point in $AB$ is also in $C$. But I have difficulty in proving it. The case $ABCO$ is colinear is easy, where $O$ is the centre…
bon li
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Get canonical equation of ellipse

We have an ellipse with a circle in it. The circle is passing through the two vertices and through the ellipse's center. It's diameter equals 7. We have also an equilateral triangle which vertices are in ellipse's focuses and in the minor vertex…
ivkremer
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Incidence Geometry

Consider a quadraple $(a,b,c,d)$ of points in the real plane such that $|ab| = |cd|$. If the perpendicular bisector of line segment $ac$ is parallel to perpendicular bisector of $bd$, then how does one see that there exists a unique translation that…
Jones
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Determining the distance between 2 points based on their know distances to several other points.

I have 2 points in 3-dimensional space. I need to know the distance between them. The problem is, I do not know their coordinates. The only thing I have to go on are 6 other points in this space. The distance to these points from both the points in…
Koen027
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Triangle orthocenter problem

I found a theorem written in a clumsy way. Is this theorem true? Let $ABC$ be a triangle and $DEF$ triangle made by the base points of altitudes of $ABC$. Then the center of an incircle of $DEF$ is an orthocenter of $ABC$. And does this holds if…
geometrystudent
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The relation of angle between two slant faces of a pyramid and the angles between slant vectors

Have any of you seen this theorem of relationship of the angles between two slant faces of a pyramid and the angles between slant vectors, provided that two faces of corresponding to $\phi$ and $\eta$ are perpendicular? I attach the picture The…
Daniel
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Finding an angle between the side of a triangle and a segment from a point inside the triangle.

Question given below: ABC is a triangle and D is a point inside ABC such that: $$ m(\widehat{DCB})=m(\widehat{CBD})=18^{\circ}\\ m(\widehat{ACD})=24^{\circ}\\ m(\widehat{DBA})=12^{\circ}\\ m(\widehat{DAC})=\alpha=? $$ This is supposed to be a…
Alistair
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Proof of a point beween two different points. (Geometry)

I'm struggling with some of the logic writing this proof. This is the question: Prove that if X is in AB (AB is a line segment) with X =/= B, then dist(AX) < dist(AB). Logically this makes perfect sense. The problem is that I struggle with putting…
Milton Green
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Angles in a circle

I have troubles to prove the following: Let $\Gamma$ be a circle with center $O$, $a$ be a tangent to $\Gamma$, $A=a\cap \Gamma$, $D$ a point on $a$ and $B\in \Gamma$ such that $D$ and $B$ lies on the same side of the line $OA$. Prove that $\angle…
geometrystudent
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