Two concentric circles of radius R and r, with R>r are both intersected by the same secant line. The points of intersection, in order, are A,B,C,D.
Prove AC*CD is constant.
Align the circles so the secant is
parallel to the xaxis, then reflect the secant to create two rectangles. Let the distance from the center of the
circles to the secant be h.<o:p></o:p>
Then from the small circle BC^2 = 4r^2 –
4h^2 and from the large circle AD^2 = 4R^2 – 4h^2.
Subtracting and factoring gives
(AD+BC)*(ADBC) = 4R^2 – 4r^2.
Now AD = AB+BC+CD and by
symmetry AB=CD so AD=2AB+BC. Then (AD+BC)=2AB+2BC
and (ADBC)=2AB, so (AB+BC)*(AB) = AC*CD = R^2 – r^2

Posted by xdog
on 20180215 12:34:16 