In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.
Other less frequently used names for this class of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral.. Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral or inscribed quadrilateral, it is preferable not to use any of the last five names.
All triangles can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to be able to have an incircle.
Special cases Examples of tangential quadrilaterals are the kites, which include the rhombi, which in turn include the squares. The kites are exactly the tangential quadrilaterals that are also orthodiagonal.. A right kite is a kite with a circumcircle. If a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral, and if it is both tangential and a trapezoid, it is called a tangential trapezoid.
Characterizations In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.
According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral:
Conversely a convex quadrilateral in which a + c = b + d must be tangential.
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.
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..
A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if.
Further, a convex quadrilateral with successive sides a, b, c, d is tangential if and only if
where Ra, Rb, Rc, Rd are the radii in the circles externally tangent to the sides a, b, c, d respectively and the extensions of the adjacent two sides for each side.
Several more characterizations are known in the four subtriangles formed by the diagonals.
Contact points and tangent lengths
The incircle is tangent to each side at one point of contact. These four points define a new quadrilateral inside of the initial quadrilateral: the contact quadrilateral, which is cyclic as it is inscribed in the initial quadrilateral's incircle.
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. From each vertex, there are two congruent tangent lengths.
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. These are also the diagonals of the contact quadrilateral.
Area
Non-trigonometric formulas The area K of a tangential quadrilateral is given by
where s is the semiperimeter and r is the inradius. Another formula is
which gives the area in terms of the diagonals p, q and the sides a, b, c, d of the tangential quadrilateral.
The area can also be expressed in terms of just the four tangent lengths. If these are e, f, g, h, then the tangential quadrilateral has the area
Furthermore, the area of a tangential quadrilateral can be expressed in terms of the sides a, b, c, d and the successive tangent lengths e, f, g, h as
Since eg = fh if and only if the tangential quadrilateral is also cyclic and hence bicentric, this shows that the maximal area occurs if and only if the tangential quadrilateral is bicentric.
Trigonometric formulas A trigonometric formula for the area in terms of the sides a, b, c, d and two opposite angles is.
For given side lengths, the area is maximum when the quadrilateral is also cyclic and hence a bicentric quadrilateral. Then since opposite angles are supplementary angles. This can be proved in another way using calculus..
Another formula for the area of a tangential quadrilateral ABCD that involves two opposite angles is
where I is the incenter.
In fact, the area can be expressed in terms of just two adjacent sides and two opposite angles as
Still another area formula is.
where θ is either of the angles between the diagonals. This formula cannot be used when the tangential quadrilateral is a kite, since then θ is 90° and the tangent function is not defined.
Inequalities As indirectly noted above, the area of a tangential quadrilateral with sides a, b, c, d satisfies
with equality if and only if it is a bicentric quadrilateral.
According to T. A. Ivanova (1976), the semiperimeter s of a tangential quadrilateral satisfies
where r is the inradius. There is equality if and only if the quadrilateral is a square. This means that for the area K = rs, there is the inequality
with equality if and only if the tangential quadrilateral is a square.
Partition properties
The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites.
If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, then that line passes through the incenter.
Inradius The inradius in a tangential quadrilateral with consecutive sides a, b, c, d is given by
where K is the area of the quadrilateral and s is its semiperimeter. For a tangential quadrilateral with given sides, the inradius is maximum when the quadrilateral is also cyclic (and hence a bicentric quadrilateral).
In terms of the tangent lengths, the incircle has radius..
The inradius can also be expressed in terms of the distances from the incenter I to the vertices of the tangential quadrilateral ABCD. If u = AI, v = BI, x = CI and y = DI, then
where ..
If the incircles in triangles ABC, BCD, CDA, DAB have radii respectively, then the inradius of a tangential quadrilateral ABCD is given by
where .Bogomolny, Alexander (2016), An Inradii Relation in Inscriptible Quadrilateral, Cut-the-knot, .
Angle formulas If e, f, g and h are the tangent lengths from the vertices A, B, C and D respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ABCD, then the angles of the quadrilateral can be calculated from
The angle between the tangency chords k and l is given by
Diagonals If e, f, g and h are the tangent lengths from A, B, C and D respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ABCD, then the lengths of the diagonals p = AC and q = BD are
Tangency chords If e, f, g and h are the tangent lengths of a tangential quadrilateral, then the lengths of the tangency chords are
where the tangency chord of length k connects the sides of lengths a = e + f and c = g + h, and the one of length l connects the sides of lengths b = f + g and d = h + e. The squared ratio of the tangency chords satisfies
The two tangency chords are perpendicular if and only if the tangential quadrilateral also has a circumcircle (it is bicentric). have equal lengths if and only if the tangential quadrilateral is a kite.
The tangency chord between the sides AB and CD in a tangential quadrilateral ABCD is longer than the one between the sides BC and DA if and only if the bimedian between the sides AB and CD is shorter than the one between the sides BC and DA..
If tangential quadrilateral ABCD has tangency points W on AB and Y on CD, and if tangency chord WY intersects diagonal BD at M, then the ratio of tangent lengths equals the ratio of the segments of diagonal BD.Gutierrez, Antonio, "Circumscribed Quadrilateral, Diagonal, Chord, Proportion", , Accessed 2012-04-09.
Collinear points
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. The line containing them is the Newton line of the quadrilateral.
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..
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. Both of these lines divide the perimeter of the quadrilateral into two equal parts. More importantly, the Nagel point N, the "area centroid" G, and the incenter I are collinear in this order, and NG = 2GI. This line is called the Nagel line of a tangential quadrilateral..
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. Then the points P, HX, HY, HZ, HW are collinear.
Concurrent and perpendicular lines The two diagonals and the two tangency chords are concurrent.Yiu, Paul, Euclidean Geometry, , 1998, pp. 156–157.Grinberg, Darij, Circumscribed quadrilaterals revisited, 2008 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. But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same argument with the other two points of tangency completes the proof of the result.
If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the diagonals intersect at P, then JK is perpendicular to the extension of IP where I is the incenter.
Incenter The incenter of a tangential quadrilateral lies on its Newton line (which connects the midpoints of the diagonals)..
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