Apollonian

Concerning the Apollonian circle

Given triangle ABC, its Apollonian circle (EC) (locus of points C*, such that C*A/C*B = CA/CB) , its circumcircle (DA) and circle (HA) passing through D, A, B. Show that points C, G, I (I diametral of D w.r. to H) are on a line.

The Apollonian circle (EC) cuts orthogonally the circumcircle (DA). Points A, B are inverse w.r. to circle (EC). Hence every circle passing through these points will cut circle (EC) also orthogonally. Especially circle (IA) will cut orthogonally both (EC) and (DA). Hence I will have equal powers to these circles (equal to the square of IB). Hence I will be on the radical axis (CG) of the two circles (EC) and (DA).For a nice application of this result look at the file: TriangleSimilaritiesProd.html .

Concerning the Apollonian circle

See Also