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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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Solving the following relation in triangle

If a line through the centroid $G$ of a triangle ABC meets $AB$ in $M$ and $AC$ on $N$ then prove that $AN. MB+AM. NC=AM. AN$ both in magnitude as well as sign I tired dividing the equation by $AM. AN$ Thus resultant became…
Apoorv Jain
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Angle $x\widehat{O}y $ and a point A inside it. Is it true that $d(A,Ox)*d(A,Oy)=c$

Let there be an angle $x\widehat{O}y $ and A a random point inside it(excluding the rays Ox and Oy). Is it true that the product $d(A,Ox)*d(A,Oy)$ is constant regarless of A? If so, provide the proof (or at least a hint of it)
nick
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A mixtilinear tangency

Let ABC be a triangle with incircle $\gamma$ and circumcircle $\Gamma$. Let $\Omega$ be the circle tangent to rays $AB, AC,$ and to $\Gamma$ externally, and let $A^{\prime}$ be the tangency point of $\Omega$ with $\Gamma$. Let the tangents from…
shadow10
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Square inscribed in a circle and an angle

Consider this badly drawn picture and the circle in the picture. Suppose that the circle has unit length, so that the area is $\pi$. Suppose that We know that the area enclosed by $ABCD$ is exactly $\pi/12$. Can one then find the angle $CAB$?If so,…
MichaelManning
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Solve the following problem....

My problem is: In a circle of radius $R$ is inscribed an equilateral triangle $ABC$. Through the point $C$ is drawn a line which intersects $AB$ in point $M$ and the circle, for the second time, in point $N$. Determine $CM\cdot CN$. My idea is…
wonderingdev
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how to find three vertices of a triangle.

Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle. How can can I determine the location of the three vertices of the triangle?
user137700
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Basic Euclidean Geometry, Circle Arc

So, here is my problem explained as best as I can. I'm working on some navigation logic for a wheeled vehicle, but I've not the foggiest idea of how to do much path finding, really. So, my basic idea is that I have 2-3 points (starting and ending,…
CamelopardalisRex
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Quadrilateral problem

Assume a quadrilateral $ABCD$ and $M, N$ points on $AB$ and $CD$ respectively, such as $\frac{AM}{MB}=\frac{CN}{ND}$. Lines $AN$ and $MD$ intersect on $K$ and lines $MC$ and $BN$ intersect on $L$. Prove that the area $(KMLN)$ equals to the sum of…
jacie
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Tough geometry question

I've been working on this problem for a while but I can't seem to figure it out, so any explanations regarding how to solve it would be appreciated. Here it is: Let $AB$, $CD$, and $EF $be three parallel chords that are non-diameters of a circle on…
John
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Proving a triangle is isoceles

In the graphic we have an isosceles triangle, and the problem is Calculate $\text{m}\angle BCD$ I added the point $E$ at distance $x$ from $C$ because it causes $DE=x$, after playing with geogebra. With this, the question is easily solved. Of…
chubakueno
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Properties of a Square

So I have that squares A and B are congruent and one vertex of B is at the center of A. The question is what is the ratio of the shaded area to the area of square A. My question is if two square are congruent then must their sides also be the same?
Gamecocks99
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Calculating the Euclidean Distance

How do I calculate the Euclidean Distance between $(22, 1, 42, 10)$ and $(20, 0, 36, 8)$? I think I know the formula, but I don't know how to apply it to this data. $$d(p, q) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2 + \ldots + (q_n-p_n)^2}$$ I don't…
Mike John
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Calculating the volume of a $4$-simplex with edges of length $1$

In the Euclidean space $(\mathbb{E}^{4})$, consider a $4$-simplex $S$ with vertices $P_{0}, P_{1}, P_{2}, P_{3}, P_{4}$. Assume that the edges of the simplex $S$ have a length of $1$. (a) Calculate the volume of the simplex $S$. (b) Determine the…
Hoc Toan
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A picture geometry problem

My approach: I know only one way to relate inradius and sides of a triangle which is $\text{Area of triangle = (inradius)(semi-perimeter)}$ I am trying to get $RS$ and altitude of $\Delta AMB$ in terms of $r$ which equals to $b$ and $a/2 \; $…
user1150809
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Prove ratio $QS:SP''$ is constant, $Q$ is inversion of $P$

I'm interested in the following problem: $S$ and $S'$ are fixed points, and $L$ is a fixed line. For every point $P$ in the plane, let the line $S'P$ intersect the line $L$ at $Z$, and let $P'$ be the intersection of the line through $S'$ parallel…
hbghlyj
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