Circle_Tangents_Properties

Circle tangents property

Consider a circle c, a tangent to it L₀ at its point C and a parallel L₁ to L₀ not intersecting c. From a point H on L₁ draw the tangents to c which
intersect L₀ at points {K,L}. Then CL*CK is constant.

This is an interesting configuration and the problem at hand gives the opportunity to list some of its properties.
(1) The circle c' through the center A of the given circle c and points {K,L} has its center P on the line AH.
(2) The other intersection points {Q,R} of the tangents {HL,HK} with circle c' define line QR symmetric to L₀ w.r. to line AH.
(3) The quadrangle LRKQ inscribed in c' is an isosceles trapezium.
(4) Points (A,M,O,H) = -1 define a harmonic division. Here M is the diametral of A and O the intersection of {L₀, QR}.
(5) Point M, defined above, is an excenter of triangle ALK.
(6) Points (A,N,C,F) = -1 define also a harmonic division, where N is the other intersection point with c' of AC and F is the intersection of {AC, L₁}.

(1) is seen by drawing first the medial lines of segments AL, AK which meet at P and define there the circumcenter of ALK. Their parallels from {L,K}
respectively meet at the diametral M of A on the circumcircle c' of ALK. Since they are orthogonal to the bisectors of triangle HLK they are external bisectors
of its angles and define an excenter of triangle HLK.
(2), (3), (5) are immediate consequences.
(4) follows from the standard way to construct the polar of a point H with respect to a circle c' (see Polar2.html ).
(6) follows from (4) and the parallelity of lines {L₀, L₁, CO, NM}.
The initial claim is a consequence of (6). This claim is also equivalent to the orthogonality of circle c' to the circle with diameter FC.

Circle tangents property

See Also