Tangential Quadrilateral - Characterizations in The Four Subtriangles

Characterizations in The Four Subtriangles

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. Chao and Simeonov proved that the quadrilateral is tangential if and only if

This characterization had already been proved five years earlier by Vaynshtejn. In the solution to his problem, a similar characterization was given by Vasilyev and Senderov. If h₁, h₂, h₃, and h₄ denote the altitudes in the same four triangles (from the diagonal intersection to the sides of the quadrilateral), then the quadrilateral is tangential if and only if

Another similar characterization concerns the exradii r_a, r_b, r_c, and r_d in the same four triangles (the four excircles are each tangent to one side of the quadrilateral and the extensions of its diagonals). A quadrilateral is tangential if and only if

If R₁, R₂, R₃, and R₄ denote the radii in the circumcircles of triangles APB, BPC, CPD, and DPA respectively, then the quadrilateral ABCD is tangential if and only if

In 1996, Vaynshtejn was probably the first to prove another beautiful characterization of tangential quadrilaterals, that has later appeared in several magazines and websites. It states that when a convex quadrilateral is divided into four nonoverlapping triangles by its two diagonals, then the incenters of the four triangles are concyclic if and only if the quadrilateral is tangential. In fact, the incenters form an orthodiagonal cyclic quadrilateral. 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). Thus a convex quadrilateral is tangential if and only if the excenters in these four excircles are the vertices of a cyclic quadrilateral.

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. If R_a, R_b, R_c, and R_d are the exradii in the triangles APB, BPC, CPD, and DPA respectively opposite the vertices B and D, then another condition is that the quadrilateral is tangential if and only if

Further, a convex quadrilateral ABCD with diagonals intersecting at P is tangential if and only if

where ∆(APB) is the area of triangle APB.

Denote the segments that the diagonal intersection P divides diagonal AC into as AP = p₁ and PC = p₂, and similarly P divides diagonal BD into segments BP = q₁ and PD = q₂. Then the quadrilateral is tangential if and only if any one of the following equalities are true:

or

or