000RM.dvi

26.5 Apollonius circle: the circular hull of the excircles 717

Appendix: Three mutually orthogonal circles with given centers

Given three pointsA,B,Cthat form an acute-angled triangle, construct
three circles with these points as centers that are mutually orthogonal to
each other.

A

B C

Y

X

Z

F H

D

E

Solution

LetBC=a,CA=b, andAB=c. If these circles have radiira,rb,rc
respectively, then

rb^2 +rc^2 =a^2 ,r^2 c+ra^2 =b^2 ,ra^2 +r^2 b=c^2.

From these,

ra^2 =

1

2

(b^2 +c^2 −a^2 ),rb^2 =

1

2

(c^2 +a^2 −b^2 ),r^2 c=

1

2

(a^2 +b^2 −c^2 ).

These are all positive sinceABCis an acute triangle. Consider the
perpendicular footEofBonAC. Note thatAE = ccosA, so that
ra^2 =^12 (b^2 +c^2 −a^2 )=bccosA=AC·AE. It follows if we extendBE
to intersect atYthe semicircle constructed externally on the sideACas
diameter, then,AY^2 =AC·AE=r^2 a. Therefore we have the follow-
ing simple construction of these circles. (1) With each side as diameter,
construct a semicircle externally of the triangle. (2) Extend the altitudes
of the triangle to intersect the semicircles on the same side. Label these
X,Y,Zon the semicircles onBC,CA,ABrespectively. These satify
AY =AZ,BZ=BX, andCX=CY. (3) The circlesA(Y),B(Z)
andC(X)are mutually orthogonal to each other.