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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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Distance between parallel lines inside of a rectangle

I have 2 parallel lines which are touching a rectangle. I know the coordinates (x1,y1), (x2,y2), (x3,y3), (x4,y4) How can I find from that the orthogonal distance between the lines? my image
lio
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Euclidean geometry exercise

I would like some help to solve this: Consider a triangle $\triangle ABC$ with $\angle A$ a right angle and $BC=20$. Divide $BC$ into four congruent segments, that is, take the points $P,Q,R\in BC$ such that $BP=PQ=QR=RC=5$. Then, compute …
Sigur
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Math problem- geometry- arbitrary points

an arbitrary point P is chosen on side BC of triangle ABC and perpendiculars PU and PV are drawn from P to other two sides of the triangle. (It may be that U or V lies on an extension of AB or AC and not on the actual side of the triangle. This…
user6422
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Find the geometric location of point $K$ in the following figure

In the equilateral triangle $ABC$, points $D,E$ move on extensions of $AB$ and $AC$ so that $BD.CE=BC^2$.Call intesection of $CD,BE$ as $K$.Find the geometric path that $K$ traverses. I tried to use notion of power of a point in order to solve…
Hamid Reza Ebrahimi
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Using Ceva's Theorem Proof on Area of a Triangle

I am having trouble identifying the height of each triangle.
Lily
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Geometric configuration with lots of properties

See. To sum up: ABCD lie on a circle m with center M. BDEF lie on a circle n with center N. ABE,ADF,BCF,CDE are collinear. (So when constructing, draw the two circles m,n, find their intersections B,D, choose some C on m, find the intersection E,F…
Hauke Reddmann
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In the following figure prove that: $OA\bot B'C'$

In the following figure prove that: $OA$ is perpendicular to $B'C'$ ($ABC$ is an arbitrary triangle,$BB'$ and $CC'$ are heights and $O$ is circumcenter) I noticed that $B,C',B',C$ are concyclic, but how does this fact help?(another set of 4…
Hamid Reza Ebrahimi
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Prove that $\angle FME=90$ (WITHOUT using triangle identity) and show that $ME=MF$

Prove that $\angle FME=90$ ( DO NOT use triangle identity) and show that $ME=MF$ (Triangle $ABC$ is right at $A$ and $AB=AC$ and $M$ is at the middle of $BC$,also $D$ is an arbitrary point on $BC$ and $DE$ and $DF$ are perpendicular to $AB$ and…
Hamid Reza Ebrahimi
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Euclidean Geometry Circles

In the diagram below, XY is a chord of the circle that goes through X, Y and Z. WZ is a tangent to the circle. XZ = YZ XZW = x-30 degrees XZY = 4x I want to calculate the value of x in order to move on to the next steps. Any advice or tips on where…
Nicole Carr
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Find radius of a circle using stewart theorem

A circle C of radius 5 cm and two circles C1 and C2 of radius 3,2 respectively . C1C2 touch each other externally and both touch C internally . A circle C3 touch C1,C2 externally and touch C internally of radius r . We have to find radius r . , i…
Koolman
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Piecewise linear curve where the closest vertex always belongs to closest edge

Take a piecewise linear curve $L$ in Euclidian space, i.e. a an ordered set of points $P$ sequentially connected by straight lines $l_{i}$, each defined by two points $p_i$ and $ p_{i+1}$. Some such lines $L$ have the property that for any point…
Erik
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Analytical Geometry medial triangle

The median $AB_1$ meets the side $A_1C_1$ of the medial triangle $A_1B_1C_1$ and $CP$ meets $AB$ in $Q$ show that $AB=3AQ$. I tried to use Ceva's theorem but couldn't do that as according to Ceva's theorem if we consider P to be the meeting point…
Harsh Sharma
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Draw polynomials to demonstrate Euclid's axioms.

I've a problem with Euclid's axioms. I understand them, but now I want some equations (polynomials) that I can use to draw some graphics and probe these axioms. For example, a rect equation that probes Euclid's third axioms: "To describe a circle…
Laerion
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Find the relation (above/below) a plane and a line

$l: x=0,y=t,z=t$ and $\pi:6x+2y-2z=3$ find if they are parallel and how is above the other. So I took the dot product $(0,1,1)\cdot(6,2,-2)=0$ so they are parallel. To test how is above/below I have set $x=0$ and looked at the $z$ component. for…
gbox
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Prove that if $A \perp B$ and $B \perp C$, then $A \parallel C$

Suppose that $A$, $B$ and $C$ are non-zero vectors in $\mathbb R^2$. Show that if $A$ and $B$ are orthogonal and $B$ and $C$ are orthogonal then $A$ and $C$ are parallel. I feel like this should be very simple. My initial thought was to do a proof…
User123454321
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