criterion for a tangential quadrilateral to be also cyclic, (AM)(CP)=(BN)(DQ

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criterion for a tangential quadrilateral to be also cyclic, (AM)(CP)=(BN)(DQ

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#1 h May 16, 2019, 3:12 PM • 3 Y

Y by kiyoras_2001, Adventure10, Mango247

In a quadrilateral $ABCD$ , a circle is inscribed that is tangent to the sides $AB, BC, CD$ and $DA$ at points $M, N, P$ and $Q$ , respectively. If $(AM) (CP) = (BN) (DQ)$ , prove that $ABCD$ is an cyclic quadrilateral.

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#2 h May 21, 2019, 10:25 AM • 2 Y

Y by Adventure10, Mango247

Let $Z$ be the Miquel point associated to the configuration of quadrilateral $ABCD$ together with the diagonal points $X=AB\cap CD$ and $Y=AD\cap BC$ . So $Z$ is the center of a spiral similarity $f_1$ sending $A\to D$ and $B\to C$ , and also $Z$ is the center of another similarity $f_2$ sending $A\to B$ and $D\to C$ .

The condition $\frac{AM}{BM}=\frac{DP}{CP}$ implies that $f_1(M)=P$ , therefore by the theory of Miquel points it follows that the circle $\odot(XMP)$ passes through $Z$ , but this circle has diameter $XI$ , so $\angle XZI=90^o$ .

Similarly, from $\frac{AQ}{DQ}=\frac{BN}{CN}$ it follows that $f_2(Q)=N$ , therefore the circle $\odot(YQN)$ (which has diameter $YI$ ), also passes through $Z$ , which implies $\angle YZI=90^o$ , and in particular $Z$ lies on the line $XY$ , a condition that characterizes the cyclicity of $ABCD$ .

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