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


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



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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