POLYA015:  Circle tangent to an ellipse contained in a rectangle (proposed by Paul Yiu, 11/26/01).

An ellipse is tangent to a rectangle at the midpoints of its four sides. Give a ruler-and-compass construction of the circle tangent to the ellipse and two adjacent sides of the rectangle.

Discussion. [JPE, 11/28/01]: Choose rectangular coordinates where the ellipse is e(x,y)=x^2/a^2+y^2/b^2 -1=0
Put A=(a,0); B=(0,b); S=(a,b). Consider a circle tangent to SA, SB with radius r : c(x,y) = (x-a+r)^2+(y-b+r)^2-r^2=0
Note that (1) : the conic c(x,y) - a b e(x,y) = 0 degenerates in two lines D1, D2 with pentes (+/-) root(b/a).
Hence, if we want, for instance, the circle externally tangent to the ellipse, the common tangent is parallel to y = - root(b/a) x, which means
that the contact point M is [ a cos(u), b sin(u) ] where tan(u) = root(b/a). Now, if S' = (a+b,0), the circle with diameter OS' meets the half-line AS at U; the half-line OU meets the circle (O,OA) at P. we have tan (AOP) = root(b/a). Hence the affinity (x,y)->(x,bx/a) will map P to M. More precisely, if C = (0,a), the line PC intersects OA at P', the perpendicular from P to OA intersects P'B at M. If the perpendicular at P to OP intersects OA at P", MP" is the common tangent.

Note that we can find a construction of the common points of the ellipse with any circle c(x,y) = 0 - even not tangent to the ellipse - The directions of D1, D2 are the direction of OP and of the reflection of OP wrt OA. Moreover, if W is the center of the circle, Wx, Wy the projections of W on OA, OB, the common point of the lines D1, D2 is the common point of the lines AB and WxWy. This way, we construct D1, D2 and their common points with the circle are the required points.

[PY, 11/29/01]:  The choice of  k = ab to split c(x,y) - k e(x,y) follows from

|1-k/a^2            0                       -(a-r)             |
|   0               1-k/b^2                 -(b-r)             |  = 0.
| -(a-r)            -(b-r)     (a-r)^2+(b-r)^2-r^2+k|

Here is a factorization of c(x,y) - ab e(x,y):
D1:  (sqrt a + sqrt b)(x. sqrt b + y.sqrt a) - sqrt(ab). (a+b+sqrt(ab)-r) = 0,
D2:  (sqrt a - sqrt b)(x. sqrt b -  y.sqrt a) - sqrt(ab). (a+b-sqrt(ab)-r)  = 0.
The radius of the circle tangent to the ellipse (and the line D1) is r = (sqrt(a+b) - sqrt a)(sqrt(a+b) - sqrt b).

Bibliography.