Reformulate the problem a little: (O), (P) are circles with diameters AC > AB tangent at A. Perpendicular bisector of BC cuts (O) at E, F, so that triangle BEC is isosceles (or BECF is a rhombus). Circle (Q) is internally / externally tangent to (O), (P) and to BE. Prove BQ is perpendicular to AC.

EB, FB cut (O) again at K, L. FE bisects angle ∠CFL = ∠CFB, implying EL = EC = EB so that B is the incenter of ΔCKL. Perpendicular to AC at B cuts (O) at M, N. By construction, (O) cuts circle (B) with center B and diameter MN in diametrically opposite points which means that the inversion in (B) with negative power -BM² takes S into itself. Line EBK through the inversion center also goes to itself. Power of B to (O) is -BM² = BM × BN = BA × BC, meaning that the inversion takes A to C. Circle (P) with diameter AB goes to line perpendicular to AC at C, the external bisector of the angle ∠KCL. Let I_k, I_l be K-, L-excenters of ΔCKL. The inverse image (Q') of (Q) is tangent to I_kI_l, to BE = BI_k, and (by Feuerbach's Theorem) externally tangent to the 9-point circle (O) of the triangle BI_kI_l, therefore (Q') is its excircle in the angle BI_kI_l, centered on the external bisector of the angle EBF, i.e., the angle I_kBI_l, perpendicular to the internal bisector BC of this angle, therefore BQQ' is perpendicular to AC.

This permits generalization to a non-isosceles triangle CKL with circumcircle (O) and incenter B and angle bisector CBA. (P) is circle with diameter AB, (Q) is circle internally tangent to (O), externally tangent to (P), and to the angle bisector KBE. No change in the proof that (O) is also tangent to the other angle bisector LBF.

Another solution makes use of the common inversion.

Inversion - Introduction

• Angle Preservation Property
• Apollonian Circles Theorem
• Archimedes' Twin Circles and a Brother
• Bisectal Circle
• Chain of Inscribed Circles
• Circle Inscribed in a Circular Segment
• Circle Inversion: Reflection in a Circle
• Circle Inversion Tool
• Feuerbach's Theorem: a Proof
• Four Touching Circles
• Hart's Inversor
• Inversion in the Incircle
• Inversion with a Negative Power
• Miquel's Theorem for Circles
• Peaucellier Linkage
• Polar Circle
• Poles and Polars
• Radical Axis of Circles Inscribed in a Circular Segment
• Steiner's porism
• Stereographic Projection and Inversion
• Tangent Circles and an Isosceles Triangle
• Tangent Circles and an Isosceles Triangle II
• Three Tangents, Three Secants

Copyright © 1996-2010 Alexander Bogomolny