Metric Properties of The Incenter
The ratio of two opposite sides in a tangential quadrilateral can be expressed in terms of the distances between the incenter I and the vertices according to
The product of two adjacent sides in a tangential quadrilateral ABCD with incenter I satisfies
If I is the incenter of a tangential quadrilateral ABCD, then
The incenter I in a tangential quadrilateral ABCD coincides with the "vertex centroid" of the quadrilateral if and only if
If Mp and Mq are the midpoints of the diagonals AC and BD respectively in a tangential quadrilateral ABCD with incenter I, then
where e, f, g and h are the tangent lengths at A, B, C and D respectively. Combining the first equality with a previous property, the "vertex centroid" of the tangential quadrilateral coincides with the incenter if and only if the incenter is the midpoint of the line segment connecting the midpoints of the diagonals.
If a four-bar linkage is made in the form of a tangential quadrilateral, then it will remain tangential no matter how the linkage is flexed, provided the quadrilateral remains convex. (Thus, for example, if a square is deformed into a rhombus it remains tangential, though to a smaller incircle). If one side is held in a fixed position, then as the quadrilateral is flexed, the incenter traces out a circle of radius where a,b,c,d are the sides in sequence and s is the semiperimeter.
Read more about this topic: Tangential Quadrilateral
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“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)