Tangential quadrilateral
The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to be able to have an incircle.
In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle.
[1]: p.65 [4] If opposite sides in a convex quadrilateral ABCD (that is not a trapezoid) intersect at E and F, then it is tangential if and only if either of[4]
or Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if and only if the incircles in the two triangles ABC and ADC are tangent to each other.
[1]: p.66 A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu.
The eight tangent lengths (e, f, g, h in the figure to the right) of a tangential quadrilateral are the line segments from a vertex to the points of contact.
The two tangency chords (k and l in the figure) of a tangential quadrilateral are the line segments that connect contact points on opposite sides.
Another formula is[7] which gives the area in terms of the diagonals p, q and the sides a, b, c, d of the tangential quadrilateral.
[12] Another formula for the area of a tangential quadrilateral ABCD that involves two opposite angles is[10]: p.19 where I is the incenter.
This formula cannot be used when the tangential quadrilateral is a kite, since then θ is 90° and the tangent function is not defined.
[13] This means that for the area K = rs, there is the inequality with equality if and only if the tangential quadrilateral is a square.
[15] If the incircles in triangles ABC, BCD, CDA, DAB have radii
[19] If M1 and M2 are the midpoints of the diagonals AC and BD respectively in a tangential quadrilateral ABCD with incenter I, and if the pairs of opposite sides meet at J and K with M3 being the midpoint of JK, then the points M3, M1, I, and M2 are collinear.
If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the extensions of opposite sides in its contact quadrilateral intersect at L and M, then the four points J, L, K and M are collinear.
[20]: Cor.3 If the incircle is tangent to the sides AB, BC, CD, DA at T1, T2, T3, T4 respectively, and if N1, N2, N3, N4 are the isotomic conjugates of these points with respect to the corresponding sides (that is, AT1 = BN1 and so on), then the Nagel point of the tangential quadrilateral is defined as the intersection of the lines N1N3 and N2N4.
More importantly, the Nagel point N, the "area centroid" G, and the incenter I are collinear in this order, and NG = 2GI.
[21] In a tangential quadrilateral ABCD with incenter I and where the diagonals intersect at P, let HX, HY, HZ, HW be the orthocenters of triangles AIB, BIC, CID, DIA.
[11][10]: p.11 One way to see this is as a limiting case of Brianchon's theorem, which states that a hexagon all of whose sides are tangent to a single conic section has three diagonals that meet at a point.
From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point.
Repeating this same argument with the other two points of tangency completes the proof of the result.
[20]: Cor.4 The incenter of a tangential quadrilateral lies on its Newton line (which connects the midpoints of the diagonals).[22]: Thm.
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.
[25][26] (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
In the nonoverlapping triangles APB, BPC, CPD, DPA formed by the diagonals in a convex quadrilateral ABCD, where the diagonals intersect at P, there are the following characterizations of tangential quadrilaterals.
Let r1, r2, r3, and r4 denote the radii of the incircles in the four triangles APB, BPC, CPD, and DPA respectively.
23–24 In 1996, Vaynshtejn was probably the first to prove another beautiful characterization of tangential quadrilaterals, that has later appeared in several magazines and websites.[1]: pp.
[1]: p.74 A related result is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of its diagonals).
73 A convex quadrilateral ABCD, with diagonals intersecting at P, is tangential if and only if the four excenters in triangles APB, BPC, CPD, and DPA opposite the vertices B and D are concyclic.[1]: p.
