Triangles (5) ... Excircles (or Escribed circles) + Bevan Circle, Gergonne Point and Nagel point

 

An excircle (or escribed circle) of a triangle is a circle that is tangential to one side of the triangle and the other two sides extended. 

There are three such circles, one corresponding to each side of the triangle. 

In the diagram above the circles with centres X, Y, Z are the excircles of △ ABC

The centre of each such circle, an excentre of the triangle, is at the intersection of the bisector of the angle opposite to the side tangential to the circle and the external bisectors of the other two angles of the triangle. 

In the above diagram, the centre of the excircle with centre X is at the intersection of the bisector of $$\angle$$ABD with the bisector of $$\angle$$ACB. 

*** TRIANGLE XYZ ***

$$\angle \; ABD\; =$$$$\angle \; CBE\; ....\; (vertically\; opposite\; angles)$$

$$BX\; bisects$$$$\angle \; ABD\; and\; BY\; bisects$$$$\angle \; CBE$$

$$Hence\; XBY\; is\; a\; straight\; line.$$

$$Similarly,\; YCZ\; and\; ZAX\; are\; straight\; lines.\; It\; follows\; that\; XYZ\; is\; a\; triangle.$$

 △ XYZ is known as the EXCENTRAL Triangle

Note also that the centre of the INCIRCLE of  △ ABC is the point where the internal bisectors of the angles of  △ ABC intersect. In the diagram above the incircle of  △ ABC is shown in red.

*** The BEVAN circle ***

The circumcircle of the excentral triangle is known as the BEVAN circle.

The Bevan circle is related to the excentral triangle of a given triangle, and its radius is twice the circumradius (R) of the original triangle. This means the radius of the Bevan circle is 2R.

Mathworld.wolfram.com - Bevan Circle 

The centre of the Bevan Circle is known as the Bevan point - Wikipedia. 

See also the excellent Gogeometry.com - The Bevan point etc

*** The Gergonne Point ***

The Gergonne point is the concurrence point for the cevians from each vertex to the point on the opposite side where the inscribed circle is tangent.  

The concurrency can be proved using Ceva's theorem - see

Polymathematics.typepad.com - The Gergonne and Nagel points

The Gergonne point is named after Joseph-Diaz Gergonne (1771 - 1859)

*** The NAGEL point ***

The Nagel point is the concurrence point for the cevians from each vertex to the point on the opposite side where that side's escribed circle is tangent

The concurrency can be proved using Ceva's theorem - see

Polymathematics.typepad.com - The Gergonne and Nagel points

The Nagel point is named after Christian Heinrich von Nagel, a nineteenth-century German mathematician, who wrote about it in 1836. 

Reading-  

Handwiki.org - Incircle and Excircles of a triangle

Allyson Faircloth - University of Georgia - Investigating Excircles

and  Dario Gonzalez Martinez - Incircle, Excircles and Nine Point Circle