Constructions of equi-inclined lines in cyclic and tangential quadrilaterals

In a cyclic quadrilateral, the perpendicular bisectors of the sides are concurrent at the circumcentre. Similarly, in a tangential quadrilateral, the bisectors of the interior angles are concurrent at the incentre. Thus one way of thinking of possible generalisations is to take equi-inclined lines (for a more precise explanation see below) instead of perpendicular bisectors (cyclic quadrilaterals) and angle bisectors (tangential quadrilaterals). To ask and answer such ‘what if?’ questions (What if the bisectors are not perpendicular but equi-inclined? What if we take equi-inclined lines instead of angle bisectors?) is very common in the process of doing mathematics. Asking such questions is a crucial part of posing problems (see [Citation1, Citation2]), and an important resource, one could even say a ‘motor’, for providing new results and insights in mathematics. In this paper we will explore the above-mentioned ‘what if?’ questions.