Articles of quadrilateral

Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon

How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon? It’s quite easy to solve for triangles for the same question, logic remains same, we need to choose $4$ vertices with none of them being consecutive, what I did is […]

Proof of Ptolemy's inequality?

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge \overline{AC}\cdot \overline{BD}$$ I know this is a generalization of Ptolemy’s theorem, whose proof I know. But I have no idea on this one, can anyone help?

Radius of a cyclic quadrilateral given diagonals

I noticed something curious about intersecting chords in a circle. Suppose two chords have lengths $p$ and $q$ and intersect at right angles at point $O$. The intersection $O$ divides the two chords into four total segments of lengths $a,b,c,d$ (say $a+c=p$ and $b+d=q$). The radius of the circle turns out magically to be the […]

Is every parallelogram a rectangle ??

Let’s say we have a parallelogram $\text{ABCD}$. $\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \text{ADC}=ar\triangle \text{BCD}$$ Now the things those should be noticed are that: In $\triangle \text{ADC}$ and $\triangle \text{BCD}$: $$\text{AD}=\text{BC}$$ $$\text{DC}=\text{DC}$$ $$ar\triangle \text{ADC}=ar\triangle \text{BCD}.$$ Now in two different triangles, two […]

If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$

If $ABCD$ is a cyclic quadrilateral, then $$ AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) $$ I tried using many approaches, but I could not find a proper solution. Can anyone please help me with this problem?

Any employment for the Varignon parallelogram?

The midpoints of the sides of an arbitrary quadrilateral form a parallelogram, which is called the Varignon parallelogram of the quad. While answering a question about Quadrilateral Interpolation it has been found that the Varignon parallelogram may be considered as a Finite Difference five point star in disguise. Moreover it has been derived that the […]

Prove the triangle is equilateral given that a quadrilateral related to its circumcircle is a kite

Let $\triangle ABC$ be a triangle. Let $Γ$ be its circumcircle, and let $I$ be it’s incenter. Let the internal angle bisectors of $∠A,∠B,∠C$ meet $Γ$ in $A’,B’,C’$ respectively. Let $B’C’$ intersect $AA’$ at $P$, and $AC$ in $Q$. Let $BB’$ intersect $AC$ in $R$. Suppose the quadrilateral $PIRQ$ is a kite; that is, $IP […]

Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area

Trisect sides of a quadrilateral and connect the points to have nine quadrilaterals, as can be seen in the figure. Prove that the middle quadrilateral area is one ninth of the whole area.

Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the geometric figures has the greater perimeter, the square […]

Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral

I’ve been reading everything I can on the perspective mapping between a 2D rectangle and the projection onto the plane in 3D space of a rectangle. I’ve learned that any such quadrilateral resulting from the projection can be mapped to any rectangle. I’ve learned that the only constraints are that the quadrilateral must be convex. […]