Mathematics Stack Exchange
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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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Reading geometry problems.

Prove that the external bisector of an angle of a triangle (not isosceles) divides the opposite side (externally) into two segments proportional to the sides of the triangle adjacent to the angle That is, show $\dfrac{BA}{BC} =…
Lemon
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What hypothesis on r, r', and d ensures that y and y' intersect in only one point, i.e., that the circles are tangent to each other?

In the real Euclidean plane, let y be a circle with center A and radius of length r. Let y' be another circle with center A' and radius of length r', and let d be the distance from A to A' (see Figure 3.42). There is a hypothesis about the numbers…
A Mere Pigeon
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A advanced geometry question

In triangle $\triangle ABC, \overline{AB}>\overline{AC}$. Mis foot of perpendicular from $C$ on external bisector of angle $\widehat{A}$. $D$ is midpoint of $\overline{BC}$ .Prove that $\overline{AB}+\overline{AC}=2\overline{DM}$.
Kaustubh Marathe
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Triangle geometry problem

Problem: Given a right triangle $ABC$ with $A=90°$. On $AC$ we label $D$ such that $\angle ABD=(1/3)\angle ABC$. On AB we label $E$ such that $\angle ACE=(1/3)\angle ACB$. $F$ is the intersection of $BD$ and $CE$. The angle bisector of BFC and FBC…
Lê Đức Minh
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How do I prove this geometry problem?

Let $ABCD$ be a square and $M, N$ points on sides $AB, BC$, respectably, such that $\angle MDN = 45◦$ . If $ R$ is the midpoint of $MN$ show that $RP = RQ$ where $P, Q$ are the points of intersection of $AC$ with the lines $MD, ND.¢$
Archimedesprinciple
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if a and b are legs in a pythagorean triple and a=2mn and b=m^2-n^2 proof a = b ± 1 implies 2n^2 ± 1 is a perfect square.

If $m > n$ and $a = 2mn$, $b = m^2 − n^2$ and $c = m^2 + n^2$ then $(a, b, c)$ is a Pythagorean triple. Show that triples where $a = b\pm 1$ will only occur if $2n^2\pm 1$ is a perfect square. For the life of me, I can't figure it out. what am I…
Alosapien
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Alitudes Bisection of a Triangle

Given an acute triangle DFG, let A, B, and C denote the feet of the altitudes of DFG through D, F, and G. Prove that AD, BF, and CG bisect the angles of the triangle ABC.
j.stat
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Tangency of a Circle

Consider two circles which intersect at the points $B$ and $C$. From a point $A$ on one circle, the rays from $A$ through the points $B$ and $C$ intersect the second circle in the points $D$ and $E$. Prove that the tangent at $A$ is parallel to the…
j.stat
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How do you prove a triangle with the hypotenuse of length 5 and other sides with lengths 3 and 4 is a right triangle?

I can see that it obviously will be satisfied by the pythagorean theorem, that is: $$3^2+4^2=5^2$$ But I am sure this isn't the way to prove the statement since you are making the assumption that it is a right triangle. I know I also can't use any…
user3000482
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Straight line as circle in Euclidean geometry

In Euclidean geometry ,is it possible to have two concentric circles of infinite radius?
user402705
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What is the antonym of origin in a cartesian plot?

I am looking for a term which means "place at which both variables are highest on a plot. Does such a word exist?
BFH
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Euclid's Proposition 6 Book I proof by circle

Euclides proves proposition 6 in book I using a reductio ad absurdum proof assuming that line AB is less than line AC. Couldn't we just draw a circle with center A and distance B, and by definition 15 prove that AB = AC, as described in the…
felipegf
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Constructing 20 degree angle using compass

I watched in one of the lecture series on YouTube that it is not possible to construct a 20 degree angle using only a compass and ruler; is there a formal proof for this ?
user366398
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Let $I$ be the incenter of triangle $ABC$.Prove that the circumcenter of triangle $BIC$ lies just in the middle of arc $BC$ of circumcircle of $ABC$.

Let $I$ be the incenter of triangle $ABC$.Prove that the circumcenter of triangle $BIC$ lies just in the middle of arc $BC$ of circumcircle of $ABC$. Since $MB=MC$ so $M$ lies on orthogonal bisector of $BC$.Now it remains to prove $MO=BO$ or…
Hamid Reza Ebrahimi
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With a compass and straightedge, is it possible to construct the square of a given distance?

Given two line segments A and B of arbitrary length, is it possible to construct a segment C such that the proportion of C to A is equal to the proportion of [the area of a square with side B] to [the area of a square with side A]?
user8399
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