"A quadrilateral is tangential, if and only if, the incircles of the two triangles formed by a diagonal are tangent to each other".

I've taken the liberty of naming this theorem after two Croatian colleagues, Jelena Gusić & Petar Mladinić (2001), who as far as I’ve been able to ascertain, have priority in first publishing the result in a journal Poučak. Later publications by Worrall (2004) and Josefsson (2011) also mention and prove the theorem.

References
i) Gusić, J. & Mladinić, P. (2001). Tangencijalni četverokut. Poučak, No. 7, October, pp. 46-53.
ii) Josefson, M. (2011), More characterizations of tangential quadrilaterals (PDF), Forum Geometricorum, 11: 65–82.
iii) Worrall, C. (2004). A Journey with Circumscribable Quadrilaterals (PDF), Mathematics Teacher, Vol. 98, No. 3, October, pp. 192-199.

Investigate
1) In the first dynamic sketch below, a general quadrilateral ABCD is shown with diagonal BD drawn and the incircles of triangles ABD and BCD constructed. What do you notice about the distance EF?
2) Drag any of the vertices until E and F coincide and click on the Show Incircle button. What do you notice?
3) Now click on the Link to Theorem of Gusić & Mladinić and use your observations in 1) and 2) to prove it.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Theorem of Gusić & Mladinić

Further Investigation
Here's a neat application to a tangential hexagon of the general quadrilateral result giving EF as the absolute value of the difference between the two sums of opposite sides: Tangential Hexagon Incircles Application.