Deltoid, kite - information

Deltoid - is a quadrilateral whose four sides can be grouped into two pairs of equal length adjacent sides. The sides of the same length have a common vertex. Deltoid can be convex or concave. When the internal angle between the shorter sides of the deltoid is greater than 180 °, the deltoid is concave, otherwise the deltoid is convex. A concave deltoid is sometimes called an "dart" or "arrowhead" and is a kind of pseudo-triangle.

A convex deltoid has the following properties:

1. The sum of the measures of all interior angles is 2Π $$ \alpha+\beta+2\cdot\gamma=360^\circ $$
2. Formula for the shorter diagonal of the side deltoid (a) and the angle α
$$ e=a\cdot 2\sin\left(\frac{\alpha}{2}\right) $$
3. Formula for the shorter diagonal of the deltoid on the side (b) and the angle β
$$ e=b\cdot 2\sin\left(\frac{\beta}{2}\right) $$
4. Formula for the shorter diagonal of the deltoid on the sides, the longer diagonal and the angle γ
$$ e=\frac{2\cdot a\cdot b\cdot \sin\gamma}{f} $$
5. Formula for the shorter diagonal of the deltoid from the radius of the inscribed circle and the angle α i γ
$$ e=\frac{2r\cdot cos\left(\frac{\gamma+\alpha-180^{\circ}}{2}\right)}{sin\left(\frac{\gamma}{2}\right)} $$
6. Formula for the longer diagonal of the deltoid on the sides (a) (b) and angles α & β
$$ f=a\cdot cos\left(\frac{\alpha}{2}\right)+ b\cdot cos\left(\frac{\beta}{2}\right)$$
7. Formula for the longer diagonal of the deltoid on the sides (a) (b) and the shorter diagonal
$$ f=\sqrt{a^2-\left(\frac{e}{2}\right)^2}+\sqrt{b^2-\left(\frac{e}{2}\right)^2} $$
8. Formula for the longer diagonal of the deltoid on the sides, the shorter diagonal and the angle γ
$$ f=\frac{2\cdot a\cdot b\cdot \sin\gamma}{e} $$
9. Formula for the longer diagonal of the deltoid from the sides and the angle γ
$$ f=\sqrt{a^2+b^2-2\cdot a \cdot b \cdot \cos\gamma} $$
10. Formula for the longer diagonal of the deltoid on the side (a) and the angle β & γ
$$ f=\frac{a\cdot \sin\gamma}{\sin\left(\frac{\beta}{2}\right)} $$
11. Formula for the radius of the circle inscribed in the deltoid from the diagonal and the angle α & γ
$$ r=\frac{e\cdot \sin\left(\cfrac{\gamma}{2}\right)}{2\cdot\cos\left(\cfrac{\gamma+\alpha-180^\circ}{2}\right)} $$
12. Formula for the radius of the circle inscribed in the deltoid of the sides and diagonals
$$ r= \frac{e\cdot f}{2a+2b} $$
13. Formula for the area of the deltoid on the sides (a)(b) and angles α & β
$$ S=\frac{a^2\cdot \sin\alpha}{2}+\frac{b^2\cdot\sin\beta}{2} $$
14. Formula for the area of the deltoid from the sides (a)(b) and the angle γ
$$ S=a\cdot b\cdot \sin\gamma $$
15. Formula for the area of the deltoid from the diagonals
$$ S=\frac{e\cdot f}{2} $$
16. Formula for the perimeter of the deltoid on the sides
$$ L = 2a + 2b $$
17. Formula for the perimeter of a deltoid with a shorter diagonal and angle(α) & (β)
$$ L = \frac{e}{\sin\left(\frac{\beta}{2}\right)}+\frac{e}{\sin\left(\frac{\alpha}{2}\right)} $$
18. Pattern on the side (a) of the deltoid with the shorter diagonal and angle α
$$ a=\frac{e}{2\cdot\sin\left(\cfrac{\alpha}{2}\right)} $$
19. Pattern to the side of the (b) deltoid with the shorter diagonal and angle β
$$ b=\frac{e}{2\cdot\sin\left(\cfrac{\beta}{2}\right)} $$