4.10 The triangles circles

  Topography notes

Circumscribed circle (circumcircle) of a triangle

The center of a circumcircle of a triangle is given by interception of the axes of sides.

15 - circumcircle.svg

We have:  R = \dfrac{a}{2 \sin{\alpha}} = \dfrac{b}{2 \sin{\beta}} = \dfrac{c}{2 \sin{\gamma}}

If we don’t know any angles of the triangle we can use this:

 R=\dfrac{abc}{4 Area} = \dfrac{abc}{4 \sqrt{p(p-a)(p-b)(p-c)} }

Incircle, inscribed circle

210px-Incircle.svg

By Lampak, InductiveloadOwn workbazująca na / based on Image:Incircle and Excircles.svg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=5294856

The center of the inscribed ciricle of a triangle is also called incenter. The incenter is the interception of angle bisectors in a triangle.

There are these relations:

 r = \dfrac{Area}{p} = \sqrt{\dfrac{(p-a)(p-b)(p-c)} {p} }  Area = r p

Excircles,

Incircle_and_Excircles

Every triangle has three excircle, The center of an excircle is given by interception of angle bisector of internal opposite angle,  and the angle bisectors of adjacent external angles.

 Area = r_a (p-a) = r_b (p-b) = r_c (p-c)  r_a = \dfrac{Area}{p-a} = \sqrt{\dfrac{p(p-b)(p-c)}{p-a}}  r_b = \dfrac{Area}{p-b} = \sqrt{\dfrac{p(p-a)(p-c)}{p-b}}  r_c = \dfrac{Area}{p-c} = \sqrt{\dfrac{p(p-a)(p-b)}{p-c}}
Topography notes: Table of contentsby_nc_sa_2
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