Tangential Quadrilateral Theorem of Gusić & Mladinić

Jelena and Petar

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

Note: I've taken the liberty here of naming this theorem after two Croatian colleagues, Jelena Gusić & Petar Mladinić, who as far as I've been able to ascertain, seem to have priority in publishing a proof of it in the journal Poučak in 2001. 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) Josefsson, 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 button and use your observations in 1) and 2) to prove it.

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.

Learn More
To learn more about other properties of a tangential/circumscribed quadrilateral go to Logical Discovery: Tangential quadrilateral.

Published Paper
Read my paper The Tangential or Circumscribed Quadrilateral in Learning & Teaching Mathematics, Dec 2020.

Problem Application
Not surprisingly, the above theorem frequently comes up in problem solving situations. For example, consider Problem 4969 in Crux Mathematicorum, Vol. 51(2), February 2025:
Let ABCD be a circumscribed quadrilateral to a circle and let Ω₁, Ω₂ be the incircles of △ABC, respectively △ACD. Finally, let Ω₁ ∩ AB = {X}, Ω₁ ∩ BC = {Y}, Ω₂ ∩ CD = {Z} and Ω₂ ∩ AD = {T}. If XY = ZT , show that YZ // XT.
Challenge
Can you solve the problem, i.e. prove the result?
Solution
Compare your solution with Solution Problem 4969.

External Links
Tangential quadrilateral (Wikipedia)
Crux Mathematicorum (Canadian Mathematical Society)
1000 Mathematical Olympiad Problems (South African Mathematics Foundation)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Created by Michael de Villiers, 2 Sept 2020 with WebSketchpad, updated 7 March 2021; 2 Oct 2024; 18 April 2025; 2 July 2026.

Tangential Quadrilateral Theorem of Gusić & Mladinić

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Theorem of Gusić & Mladinić