Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Stewart's Theorem


Stewart's Theorem states: 
$$t^2=\frac{b^2u+c^2v}{a}-uv$$
The theorem can also be written as:
$$b^2u+c^2v=(a)(uv+t^2)$$

Proof:
In $\triangle AED$, by Pythagorean theorem, we have
$x^2+h^2=t^2$                                                                                                                                    (1)

Similarly, in $\triangle AEC$
$h^2+(v-x)^2=b^2 \Rightarrow h^2u+uv^2-2uvx+ux^2=b^2u$                                               (2)

In $\triangle AEB$, we use Pythagorean theorem again and get
$h^2+(u+x)^2=c^2 \Rightarrow h^2v+u^2v+2uvx+vx^2=c^2v$                                                  (3)

Now, adding eq. (2) and eq. (3), we get
$$b^2u+c^2v=h^2u+h^2v+uv^2+u^2v-2uvx+2uvx+ux^2+vx^2$$
$$b^2u+c^2v=(u+v)(h^2+uv+x^2)$$
$$b^2u+c^2v=(u+v)(t^2+uv)$$
$$b^2u+c^2v=(a)(t^2+uv)$$


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Equilateral and Equiangular Polygons

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