Mathematics Stack Exchange
Mathematics Stack Exchange

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.

9328 questions
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angle bisector and concurrency problem.

enter image description here In this figure, line BE is angle bisector of ∠ABC and some point X is on BE. If ∠AEB=∠FDB, then prove ∠EDF is right angle.
user613265
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Similarity of triangles

I need help with this geometric problem. $ \overline {AB}$ is a diameter in circle $M$ , $C \in$ $\overrightarrow{AB}$ lying outside the circle , $\overline{CD}$ is drawn tangent to the circle at point $D$ , then $\overrightarrow{DH} \bot…
tom fox
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Is it true that: $\| a \| \| b \| \cos \alpha = \langle a,b\rangle$

Let $a , b \in \mathbb{R}^n$ and let $\alpha$ be the angle formed between $a$ and $b$. Is it true that: $$ \| a \| \| b \| \cos \alpha = \langle a,b\rangle $$ ($\langle\cdot,\cdot\rangle$ being the dot product) If so why?
aribaldi
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rectangle of a rectangle

I need a formula for a rectangle based on a rectangle like this: Okay, so i have the black rectangle, from 0,0 to whatever xy size, and then i need a new rectangle over it (blue one) shaped based on the black one touching the angles, and to know…
Lucijan
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How can I find the diagonal of a quadrilateral?

Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?
parkhyeyoo
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Construct a regular pentagon in only 11 steps using ruler and compass.

One step is to draw a stright line or a circle (greek classical understsnding of step)
Arbusto Gonzalez Perez
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Geometric way to construct image of the image of a point bu given homothety

Let $ABC$ be a right triangle at $A$ and $G$ a point interior to it. $D$ is the where lines $(AG)$ and $(BC)$ meet. Let $h$ be the homothety of center $A$, that transforms $G$ to $D$. I wonder whether there is a geometric way to place $D'$ the image…
daiski
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Proof based on Triangles

for an acute triangle,prove that $a^2+b^2>c^2$ Well, I dropped a perpendicular from B to O but still I can't prove the question because I can't prove that $2a^2>2cz$ Where z is the side BO Pls help me
Death Champion
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How to get the distance between two anchoring points in a tv mast?

The problem is as follows: The alternatives given in my book are as follows: $\begin{array}{ll} 1.&\textrm{4 m}\\ 2.&\textrm{5 m}\\ 3.&\textrm{6 m}\\ 4.&\textrm{7 m}\\ \end{array}$ How exactly should this be solved relying only in euclidean…
Chris Steinbeck Bell
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How to find the angle by a triangle and a median and bisector?

The problem is as follows: In a triangle $\triangle{ABC}$, the measure of angle $\angle ABC = \angle 105^{\circ}$. On $AC$ it is located a point $M$, such that $AB = MC$. The mediatrices of $BC$ and $AM$ intersect at $Q$. Find the measure of $\angle…
Chris Steinbeck Bell
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Triangle, incenter and perpendicular line

For the triangle $ABC$, let $I$ the incenter and $I_A$ the A-excenter. If $L$ the midpoint of arc $BC$, we can show that $L$ is the center of a circle through $I, I_A, B, C.$ Also, if the incircle touches $AB, AC$ at $P, Q$ and $BI, CI$ intersect…
George
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An angle in a regular hexagon

The marked angle is $60^\circ$. But I wonder how the angle is found without using trigonometry. EDIT. Extra question: How can it be proved that the line DI bisects the angle GIH?
P.-S. Park
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Slope and euclidean geometry (grade 10)

Given Triangle $DEF$ with vertices $$D(-2,6),E(7,3), \text{ and } F (2,-3)$$ find: a) the equation of the altitude from vertex $E$ in standard form (include y=mx+b) b) Find where this altitude in a) intersects side $DF.$
a s
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square on the top of a square

i am stuck with a question of mensuration for three days and i posted it on facebook and asked friends but no correct solution yet. please help .
Vindhyachal Vindhyachal
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I need a help for this problem

Given an isosceles triangle. Find the locus of the points inside the triangle such that the distance from that point to the base equals to the geometric mean of the distances to the sides. Any ideas please?
user225430
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