8.5 The excenters and the excircles 237
Exercise
1.Given a triangleABC, construct a triangle whose sides have the
same lengths as the medians ofABC.2.Construct the incircle of triangleABC, and mark the points of con-
tactXonBC,Y onCA, andZonAB. Are the linesAX,BY,
CZconcurrent? If so, is their intersection the incenter of triangle
ABC?3.Given a rectangleABCD, construct pointsPonBCandQonCD
such that the triangleAP Qis equilateral.
4.LetD,E,Fbe the midpoints ofBC,CA,ABof triangleABC.
Construct the circumcircle ofDEF. This is called thenine-point
circleof triangleABC. Construct also the incircle of triangle
ABC. What do you observe about the two circles? How would
you justify your observation?5.Construct the circle through the excenters of triangleABC.Howis
its center related to the circumcenter and incenter of triangleABC?6.Given three non-collinear points as centers, construct three circles
mutually tangent to each other externally.