An excircle is a circle outside the triangle that is tangent to the three sides of the triangle.
An excenter is the center of the excircle.  For every triangle there are 3 excircles and 3 excenters.

Given triangle ABC with side lengths a, b, and c. Let circle O be an excircle and let X, Y, and Z be its points of tangency, as shown.

(a) Prove that AC + CX is the semiperimeter of triangle ABC.

Lemmas:

AY is the semiperimeter of triangle ABC.

AZ is the semiperimeter of triangle ABC.

AB + BX is the semiperimetger of triangle ABC.

(b)  Show that the excenter is the point of concurency of two adjacent external angle bisectors and the internal angle bisector of the opposite angle.

(c) Prove that triangle XYZ is obtuse, that is, that the triangle determined by the points of tangency of an excircle is always obtuse.

(d Find the area of triangle XYZ in terms of a, b, and c.

(e) Show that the lines (not segments) AX, BY, and CZ are concurrent.

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Hints/Solution:

              Click here for a GSP Sketch. Try dragging points B or C in the sketch.

Comments:

               Compare with the Incircle Problems


Extensions/Variations:

               Construct the three excircles and excenters of a triangle.  Construct the triangle with vertices at the excenters.   Observations?

                See Bisectors in a 120 Degree Obtuse Triangle problem.

            For an extension and partial solution, see Problem 4153 in School Science and Mathematics, October 1987 (??) and October 1988 issues.

           In Problem 4153, if A(I) is the area of the triangle from the tangent points of the incircle, and A(E1), A(E2), and A(E3) are the areas of the three triangles from the tangent points of the three excircles, then

Click here to see a picture. Click here for a Geometer's Sketchpad file.