# Elementary geometry questions and answers

Recent questions in Elementary geometry
Lucas Roman 2022-06-04

### Intuitionism refuses the Cantor hypothesis about continuum as a hypothesis meainingless from the intuitionistic point of view Also, 'the tertium non datur' principle:$A\vee \mathrm{¬}A$is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\mathrm{¬}A$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the 'reductio ad absurdum' to prove a theorem? Thanks

Cason Leblanc 2022-06-04

### I fooled around with the concept of an angle bisector, and I (thought I) found out (and some websites confirmed this, but now I'm in doubt) that the angle bisector of a vertex is the collection of points equidistant from the 2 sides of the vertex it's bisecting. However, how is this possible. Wouldn't this mean that the bisector would divide the opposite side in 2 equally long segments? But this would create a median, and obviously a median and an angle bisector are different things. Can anyone help?

Payton Salazar 2022-06-04

### I'm taking a course at teaching and we have some geometry questions. Among the questions there was one I couldn't solve.I'm trying to prove that angle bisectors in a triangle intersect at a single point, but I need to show it analytically, i.e using line equations and distance between points. I'm familiar with proofs using geometry and vectors, but couldn't manage to show analytically. I can't use the formula of angle between lines or distance of point from a given line.Things I've tried:1. angle bisector theorem.2. Writing equations of sides as $Ax+By+C=0$ and then creating the equation of angle bisector - quite messy and I believe there is a lot easier way.Thanks!

Estrella Le 2022-06-03

### Triangle ABC has side lengths $AB=7,BC=8,$ and $CA=9.$ Its incircle $\mathrm{\Gamma }$ meets sides BC, CA, and AB at D, E, F respectively. Let AD intersect $\mathrm{\Gamma }$ at a point $P\ne D.$ . The circle passing through A and P tangent to Γ intersects the circle passing through A and D tangent to $\mathrm{\Gamma }$ at a point $K\ne A.$. Find $\frac{KF}{KE}.$ .

Marquis Cooper 2022-06-03

### The angle bisectors $AL$ and $BQ$ in the right triangle $\mathrm{△}ABC$ with $\mathrm{\angle }ACB={90}^{\circ }$ and catheti 15 and 20 divide the median $CM$ into three segments. Find their lengths.I am not sure how to approach the problem. The most "obvious" thing that we can do is to find the hypotenuse $c=\sqrt{{a}^{2}+{b}^{2}}=25$ of the triangle $\mathrm{△}ABC$. Therefore, ${m}_{c}=\frac{1}{2}\cdot 25=12.5$. On my sketch $AC=15$and $BC=20$. What to do next?

amuguescaet3jf 2022-06-03

### In $\mathrm{△}ABC$, $BE$ and $CF$ are the angular bisectors of $\mathrm{\angle }B$ and $\mathrm{\angle }C$ meeting at $I$. Prove that $\frac{AF}{FI}=\frac{AC}{CI}$.By the angle bisector theorem we have $\frac{AC}{AE}=\frac{CB}{BE}$ and $\frac{AB}{AF}=\frac{BC}{FC}$. How do I proceed after this? Hints would be appreciated.

Kallie Arroyo 2022-06-02

### Let ABC be an acute angled triangle with circumcenter O. A circle passing through A and O intersects AB, AC at P, Q respectively. Show that the orthocentre of triangle OPQ lies on the side BC.

patzeriap0 2022-05-29

### I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more applicable in one case more than the other? I never hear of axioms in geometry or postulates in set theory. Are axioms more formal and postulates used more informally?

Alessandra Clarke 2022-05-29

### I want to show that $2{p}_{n-2}\ge {p}_{n}-1$...Bertand's postulate shows us that $4{p}_{n-2}\ge {p}_{n}$ but can we improve on this?any ideas?

Angel Malone 2022-05-29

### Hi guys I just need to know if my answer is right.The question is1) Euclids ${4}^{th}$ postulate is "That all right angles are equal to one another". Why is this not obvious?My answer:When I read this question I am like it is obvious, so I got kind of confused. But I took a crack at it anyways.If you read the postulate. This is not obvious because you dont know if the right angle is a right angle. What if a triangle was drawn differently but with one line perpendicular to another. You need to know if the line is perpendicular or not, and that is why it is not obvious.Could someone tell me if that is right or should I add in more info.

qtbabe9876a9 2022-05-28

### I don't find Postulate 13 of Spivak's Calculus trivial, nor can I understand why it's true.Postulate 13: Every non-empty set of real numbers that is bounded above has a least upper bound (sup).Why is this postulate true? Any proof/intuition behind it?Edit: Let me pose a few questions.1. Suppose I decide to devise a pathological function $\mathbb{R}$$\to$$\mathbb{R}$ which is bounded above but has no supremum. Why will I fail to find such $f$? If you simply take it as an axiom, there's no guarantee I won't be successful.2. Suppose $S$ is an arbitrary non-empty set of real numbers that is bounded above. Does there exist an algorithm to determine $sup\left(S\right)$?

res2bfitjq 2022-05-27

### How to integrate ${\int }_{0}^{\pi /2}\frac{\mathrm{sin}\left(x\right)}{\mathrm{sin}\left(x+\frac{\pi }{4}\right)}\mathrm{d}x$ using substitution $x=\frac{\pi }{2}-y$ ?

groupweird40 2022-05-26

### I kept wondering of the best way to "see" why midpoint of $A={x}_{1},B={x}_{2}$ is $\left({x}_{1}+{x}_{2}\right)/2$.

tuehanhyd8ml 2022-05-15

### What's the ratio between the segments $\frac{AF.BG}{FG}$ in the figure below?

Merati4tmjn 2022-05-15

### "The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram."I find this property very useful while solving different problems on Quadrilaterals & Polygon,so I am very inquisitive about a intuitive proof of this property.

Edith Mayer 2022-05-14

### Show that the circumscribed circle passes through the middle of the segment determined by center of the incircle and the center of an excircle.

Waylon Mcbride 2022-05-13

### Find the area of a triangle with vertices $\left(0,1,1\right),\left(-1,-1,2\right),\left(2,3,1\right)$

Jayden Mckay 2022-05-10