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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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Lines from each vertex of a triangle to diagonal intersections of constructed squares on opposite sides concur. Does this point have a name or number?

Given any triangle ABC. Join vertices A,B,C with midpoints (diagonal intersections) say P, Q, R of squares constructed on opposite sides BC, AC and AB respectively. The lines AP, BQ and CR concur at a point say H. It can be proved that this point is…
Arne Erikson
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Finding a rectangle's sides given all distances from point inside to the vertices.

Given distances from a point inside a rectangle to all the vertices, is it possible to find the sides of the rectangle, or is this analagous to the impossiblity of finding a third side of a triangle given only two of the triangle's sides?
Arne Erikson
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light geometry and art

I'm a professional artist. I have a question regarding light logic. I have simplified the problem a bit. Imagine a segment in 3D space, AB of known length and a point light illuminating the segment. Say the light is at (0,5,8) and the mid point of…
raff
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Point $D$ in $\triangle{ABC}$ with $\angle{ABD}=5^{\circ}, \angle{DBC}=20^{\circ}, \angle{DCB}=65^{\circ}, \angle{DAC}=40^{\circ}$, find $\angle{ACD}$

Point $D$ in $\triangle{ABC}$ with $\angle{ABD}=5^{\circ}, \angle{DBC}=20^{\circ}, \angle{DCB}=65^{\circ}, \angle{DAC}=40^{\circ}$, find $\angle{ACD}$. Trigonometric Ceva theorem approach is quite straight forward: $$ \begin{multline} \shoveleft…
r ne
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Prove that $BF$ bisects the line segment $DE$

Let $C$ be a point on a semicircle $\Gamma$ of diameter $AB$ and let $D$ be the midpoint of the arc $AC$. Let $E$ be the projection of $D$ onto the line $BC$ and $F$ the intersection of the line $AE$ with the semicircle. Prove that $BF$ bisects the…
PNT
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Prove that the segment $BE$ bisects $AC$

Points $A,B,C,D,E$ lie on a circle $ω$ and point $P$ lies outside the circle. The given points are such that (i) lines $(P B)$ and $(P D)$ are tangent to $ω$, (ii) $P, A, C$ are collinear, and (iii) $DE ∥ AC$. Prove that $[BE]$ bisects $[AC]$. The…
PNT
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A curious near concurrence

In a triangle $ABC$ with incircle $\omega$, excenters $I_A, I_B, I_C$, let $t_{AB}$ and $t_{AC}$ be the tangent lines to $\omega$ through $I_A$ that are closer to $B$ and $C$ respectively. Construct similarly $t_{BA}$, $t_{BC}$, $t_{CA}$,…
user68136
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Length of lines related to orthocenter

Let ABC be a triangle and let H be its orthocenter. Let M be the midpoint of BC. The perpendicular to MH through H intersects AB and AC at P and Q, respectively. Prove that |MP| = |MQ|. My attempt so far: Clearly, this can easily be solved using…
justadumbguy
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What are some benefits of learning Euclidean Geometry?

I am an undergraduate math major student. I believe Euclidean Geometry is one of the most beautiful subjects in math and helps us to improve mathematical intuition. However, there are very few universities that provide Euclidean Geometry courses.…
metsu
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colinear geometry

Let triangle $ABC$ with $(I)$ is incircle and $(I)$ tangent to $BC,CA,AB$ at $D,E,F$, respectively. Let $H\in EF$ such that $DH\perp EF$. Prove that $H$ and othorcenters of $\Delta AEF$ and $\Delta ABC$ are colinear. Here are what i have done: Let…
Dat Tran
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Looking for a Geometry Book with Exercises (Not a Text Book)

I am looking for a Geometry Book with Hard Exercises based on Euclidean Geometry that are covering topics such as Congruence, Parallelogram, Circles, Similarity, Simson Line etc... I am currently working on the Challenging Problems in Geometry Book.…
sillysillybag69
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How to formally define direction in Euclidean space?

Consider the space $\mathbb{R}^n$. A direction can be thought of as point on the unit sphere. However, I do not know how to formally define the relation "Point $x$ is more towards direction $\theta$ than point $y$", where $\theta$ is a direction.…
user107952
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Segment formed by a point inside a circle and a point outside intersects the circle?

O is the center of the circle of radius r. We have two points A and B | OA < r and OB > r. I have to prove that AB intercepts the circle. I tried to use the line segment that contains AB to build a perpendicular line that intercepts O, the point of…
Wesley
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How to find angle in between two right triangles when one side is twice of other?

The problem is as follows: $\begin{array}{ll} 1.&10^\circ\\ 2.&12^\circ\\ 3.&15^\circ\\ 4.&16^\circ\\ \end{array}$ I'm not sure which sort of construction can be used here to solve this problem?. I've attempted to draw a perpendicular line from $B$…
Chris Steinbeck Bell
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How to find the side of a triangle which is connected by two bisectors?

The problem is as follows: Using the figure from below: Find $PR$. Assume $QS=SR$ and $PQ=24\,cm$ and $TR=8\,cm$. The alternatives given in my book are as follows: $\begin{array}{ll} 1.&\textrm{16 cm}\\ 2.&\textrm{32 cm}\\ 3.&\textrm{30…
Chris Steinbeck Bell
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