Quadrilateral PQRS with diagonal PR is split into triangles PQR and PSR. Circles O1 and O2 are inscribed in PQR and PSR, touching PR at M and N respectively. Find the length MN. I. PQRS has an incircle (it is tangential). II. The inradii are r1 = 5 (for PQR) and r2 = 6 (for PSR).

Introduction / Context:
The segment MN lies on PR between the points where the two incircles touch PR. To determine MN uniquely, one would generally need sufficient constraints to locate each touchpoint along PR (i.e., their distances from P or R). We must judge sufficiency, not compute a numeric value unless forced.

Given Data / Assumptions:

• M is the touchpoint of the incircle of triangle PQR on PR; N is the touchpoint for triangle PSR on PR.
• Statement I: PQRS is tangential (has a single incircle touching all four sides).
• Statement II: Inradii r1 = 5 (PQR) and r2 = 6 (PSR).

Concept / Approach:
In a triangle, the touchpoint divides a side using the formulae based on the semiperimeter: distance from a vertex to the touchpoint equals s − adjacent side. For triangles over a shared base PR, the absolute locations of M and N depend on multiple side lengths (or equivalent data like semiperimeters and two sides). Merely knowing concentric tangency of the quadrilateral or the inradii of the sub-triangles does not by itself fix those side lengths.

Step-by-Step Solution:
With I alone: the tangential condition (sum of opposite sides equal) constrains PQRS globally (PQ + SR = QR + PS) but gives no direct lengths on PR or semiperimeters s(PQR), s(PSR); M and N remain undetermined.With II alone: the inradii r1 and r2 relate to areas and semiperimeters of the two triangles via r = A/s. Without at least one triangle’s semiperimeter or two sides, we cannot place M and N.Even I + II together leave infinitely many configurations satisfying the conditions but yielding different MN values. Hence MN is not uniquely determined.

Verification / Alternative check:
Construct non-similar examples meeting I and II that produce different distances along PR; MN varies.

Why Other Options Are Wrong:
No single statement pins the touchpoint locations; together they still do not fix PR-side splits; claiming sufficiency would be incorrect.

Common Pitfalls:
Assuming r1 and r2 fix side lengths; they do not without additional metrics such as sides or angles.

Final Answer:
Both statements together are not sufficient.