# Grid dissections of tangential quadrilaterals

## TL;DR

This paper proves that any tangential quadrilateral can be partitioned into n^2 smaller tangential quadrilaterals using an n-by-n grid dissection, extending a known property of squares to a broader class.

## Contribution

It establishes the existence of class-preserving grid dissections for tangential quadrilaterals for all integers n ≥ 2, a novel generalization of square dissections.

## Key findings

- Every tangential quadrilateral admits an n×n grid dissection into smaller tangential quadrilaterals.
- The result generalizes the checkerboard dissection property from squares to tangential quadrilaterals.
- The proof applies for all integers n ≥ 2.

## Abstract

For any integer $n\ge 2$, a square can be partitioned into $n^2$ smaller squares via a checkerboard-type dissection. Does there such a class-preserving grid dissection exist for some other types of quadrilaterals? For instance, is it true that a tangential quadrilateral can be partitioned into $n^2$ smaller tangential quadrilaterals using an $n\times n$ grid dissection? We prove that the answer is affirmative for every integer $n\ge 2$.