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The incircle-generating tetrahedron (BMT 2019 Geo #9)

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May 18, 2019
geometry3D geometrytetrahedron

Problem Statement

Let ABCD ABCD be a tetrahedron with ∠ABC=∠ABD=∠CBD=90∘ \angle ABC = \angle ABD = \angle CBD = 90^\circ and AB=BC AB = BC . Let E,F,G E, F, G be points on AD‾ \overline{AD} , BD BD , and CD‾ \overline{CD} , respectively, such that each of the quadrilaterals AEFB AEFB , BFGC BFGC , and CGEA CGEA have an inscribed circle. Let r r be the smallest real number such that [△EFG][△ABC]≤r \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r for all such configurations A,B,C,D,E,F,G A, B, C, D, E, F, G . If r r can be expressed as a−bcd \dfrac{\sqrt{a - b\sqrt{c}}}{d} where a,b,c,d a, b, c, d are positive integers with gcd⁡(a,b) \gcd(a, b) squarefree and c c squarefree, find a+b+c+d a + b + c + d .
Note: Here, [P] [P] denotes the area of polygon P P . (This wasn't in the original test; instead they used the notation area(P) \text{area}(P) , which is clear but frankly cumbersome. :P)
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