Tangent Lines and Circles in Convex QuadrilateralLet A0, A1, A2, and A3 list in order the vertices of a convex quadrilateral Q. Treat all indices as integers modulo 4. Let Lk denote the line through Ak and Ak+1, and let Ck be the circle tangent to Lk-1, Lk, and Lk+1 outside Q. Let Mk be the line through the points where Ck is tangent to Lk-1 and to Lk+1. Let Ek be the intersection of Mk with Mk+1. Prove that E1E3 bisects A1A3.
(The problem was proposed by Chu Cheng, Yan An Middle School, Shanghai, China. The solution is by Marius Stefan, Los Angeles, CA. American Mathematical Monthly, v 114 (December 2007), 926-927).  |Activities| |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2012 Alexander Bogomolny For distinct points A, B,C, write d(A, BC) for the distance from point A to the line through B and C. Denote by Tlk the point where the line Lk touches the circle Cl. Let M be the midpoint of the line segment A1A3. Let α be half the angle between L1 and L3, lines that enclose C0 and C2. Let β be half the angle between L0 and L2, the lines enclosing C1 and C3.
With these definitions, we have
Similarly,
It follows that
The lines E3E0 and E1E2 are parallel because they are both perpendicular to the bisector of 2α. The lines E0E1 and E2E3 are also parallel, since they are perpendicular to the bisector of 2β. Hence E0E1E2E3 is a parallelogram. The distance d0 between E3E0 and E1E2 is
Similarly, the distance d1 between E0E1 and E2E3 is
Therefore,
which implies that M ∈ E1E3. The configuration has additional properties:
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