Tangent ellipses

Geometry Level 5

There is a circle of a radius $\sqrt{\frac{8}{3}}$ . There are 3 ellipses inside the circle which are tangent to the circle at points $A, B, C$ and to each other at points $D, E, F$ . Their major axes are parallel to lines tangent to the circle at corresponding tangency points. The lengths of major axes of the ellipses are:
$2 \sqrt{\frac{3}{2}}$ , $2 \sqrt{\frac{8}{7}}$ and $2$ . $X = \frac{\triangle ABC}{ \triangle DEF}$

Find $\lfloor X \times 1000 \rfloor$

The answer is 3648.

1 solution

Maria Kozlowska
Jul 11, 2015

The circle with 3 inscribed tangent ellipses is just a top view of Crelle's tetrahedron incircles. In order for four circles to be tangent to each other the tetrahedron edge lengths have to meet the criteria: $a + a' = b + b'=c + c'$ where $a'$ is an edge opposite to $a$ (Crelle's tetrahedron).

Edge lengths can be found solving system of 4 equations using formula for the radius of the inscribed circle: $r=\frac{\triangle}{s}$ where s is a semiperimeter of a triangle.
In this case edge lengths are: $5,6,7,3,4,5$ , base side lengths: $5,6,7$ .

Having this information the area of the contact triangle $ABC$ can be found. Geometric construction can be made of a tetrahedron net with 4 circles. Ellipses contact points can be found as points of intersections of lines perpendicular to the sides of the base from circles contact points.

Note: This problem was inspired by Michael Mendrin's Four Tangent Circles

Tangent ellipses

The answer is 3648.

1 solution

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