163. Applications of Ptolemy's inequality. Ex. - China,1998.

by Virgil Nicula, Oct 21, 2010, 9:23 AM

PP1 (China, 1998). Let $ABC$ be an acute triangle. Ascertain the position of an inner point $D$

w.r.t. $\triangle ABC$ for which $\boxed{a\cdot DB\cdot DC+b\cdot DC\cdot DA+c\cdot DA\cdot DB=abc}$ .

Proof. Denote the points $E$ , $F$ for which the quadrilaterals $DCBE$ , $ACBF$ are parallelograms. Observe that $DE=AF=a$ ,

$BF=b$ , $BE=DC$ and $EF=DA$ because $BEF\equiv CDA$ . Apply the Ptolemy's inequality in the quadrilaterals :

$\left\|\begin{array}{cccc} ADBE\ : & AB\cdot DE\le DA\cdot BE+DB\cdot AE & \implies & ac\le DA\cdot DC+DB\cdot AE\\\\ ABEF\ : & AE\cdot BF\le AF\cdot BE+AB\cdot EF & \implies & b\cdot AE\le a\cdot DC+c\cdot DA\end{array}\right\|$ .

Thus, $abc\le b\cdot DA(b\cdot AE)\cdot DB\le b\cdot DA\cdot DC+(a\cdot DC+c\cdot DA)\cdot DB=abc$ . So the quadrilaterals $ADBE$ , $ABEF$ are cyclically $\iff$

$ABDEF$ is cyclically $\implies$ $AFED$ is rectangle $\implies$ $AD\perp BC$ . Since $BE\perp BA$ obtain $CD\perp AB$ . In conclusion, $D$ is the orthocenter of $\triangle ABC$ .

Remark. For any inner point $D$ of an acute triangle $ABC$ there is the inequality $\boxed {a\cdot DB\cdot DC+b\cdot DC\cdot DA+c\cdot DA\cdot DB\ge abc}$ .


PP2 (Cehia-Slovacia, 1998). Let $ABCDEF$ be a convex hexagon for which $AB=BC\ ,\ CD=DE\ ,\ EF=FA$ . Prove that $\boxed {\frac {BC}{BE}+\frac {DE}{DA}+\frac {FA}{FC}\ge \frac 32}$ .

Proof. Denote $AC=x$ , $CE=y$ , $EA=z$ . Apply the Ptolemy's inequality to the quadrlaterals :

$\left\|\begin{array}{cccc} ABCE\ : & x\cdot BE\le (y+z)\cdot BC & \implies & \frac {BC}{BE}\ge \frac {x}{y+z}\\\\ CDEA\ : & y\cdot DA\le (z+x)\cdot DE & \implies & \frac {DE}{DA}\ge \frac {y}{z+x}\\\\ EFAC\ : & z\cdot FC\le (y+z)\cdot FA & \implies & \frac {FA}{FC}\ge \frac {z}{x+y}\end{array}\right\|\ \bigoplus\implies$ $\frac {BC}{BE}+\frac {DE}{DA}+\frac {FA}{FC}\ge\frac {x}{y+z}+\frac {y}{z+x}+\frac {z}{x+y}\ge \frac 32$ .
This post has been edited 33 times. Last edited by Virgil Nicula, Nov 22, 2015, 8:43 PM

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