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I'm writing a ~60-page paper on cyclic, tangential and bicentric quadrilaterals. I need to give some problems (with solutions) where usage of those is "hidden". There are lots of problems that use idea of cyclic quadrilaterals and they're not problem to find, but I wasn't able to find any problems where we use idea of a tangential quadrilaterals.

Problems (with short ideas for solutions) of any difficulty are welcome, where we use ideas about tangential (or bicentric) quadrilaterals. It doesn't have to be the main idea for solution, although it's preferred.

J. M. ain't a mathematician
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Lazar Ljubenović
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2 Answers2

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Let ABCD be a quadrilateral with right angles at B and D. Let lines W and Y be perpendicular to AC through A and C respectively. Let line X through B meet lines W and Y at E and F respectively, and let line Z through D meet lines W and Y at H and G respectively. If $\angle$EBA = $\angle$ACB and $\angle$ADH = $\angle$ACD, show that EF + HG = EH + FG.

Ideas for solution: The right angles at B and D ensure that A, B, C and D lie on a circle with centre O, the mid point of AC. $\angle$EBO = $\angle$EBA + $\angle$ABO = $\angle$ACB + $\angle$BAO = $180^o$ - $\angle$ABC = $90^o$. Similarly $\angle$HDO = $90^o$. Thus EFGH is a tangential quadrilateral to the circle through A, B, C and D. Hence by a standard property of tangential quadrilaterals EF + HG = EH + FG.

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Adam Bailey
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The following was problem 1287 in Elemente der Mathematik, 66 (2011), p. 38. A solution appeared in vol. 67 (2012), 40–41.

Let $Q$ be a tangential quadrilateral. The perpendicular bisectors of its four sides bound another quadrilateral $Q'$. Prove that $Q'$ is again tangential.

A solution (in German) can be found here:

http://www.math.ethz.ch/~blatter/Tangviereck.pdf

Christian Blatter
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