Area Inequality in Triangle II

Let $ABC$ be a triangle with area $1$ and $P$ a point inside the triangle. Denote as $D,$ $E,$ $F$ the intersection points of $AP,$ $BP,$ $CP$ with the sides $BC,$ $CA,$ $AB,$ respectively.

concurrent cevians split a triangle into six parts

Prove that there are at least two triangle among $PBD,$ $PCD,$ $PCE,$ $PAE,$ $PAF$ and $PBF$ with the area at most $\displaystyle\frac{1}{6}.$

Proof

The proof has two beginnings and one end. Let $[X]$ stand for the area of of figure $X.$ Let denote by $a,$ $b,$ $c$ $[\Delta PBD],$ $[\Delta PCE],$ $[\Delta PAF],$ respectively, and by $x,$ $y,$ $z$ $[\Delta PCD],$ $[\Delta PAE],$ $[\Delta PBF].$ Let $\alpha,$ $\beta,$ $\gamma$ denote the angles around $P$ as shown below:

concurrent cevians split a triangle into six parts, solution

Then by the sine formula for the area of a triangle,

$\begin{align} a &= \frac{1}{2}PB\cdot PD\sin\alpha,\\ b &= \frac{1}{2}PC\cdot PE\sin\gamma,\\ c &= \frac{1}{2}PA\cdot PF\sin\beta, \end{align}$

so that $abc=PA\cdot PB\cdot PC\cdot PD\cdot PE\cdot PF\sin\alpha\sin\beta\sin\gamma.$ Clearly, $xyz$ equals the same quantity, implying $abc=xyz.$

The latter identity can be also derived from Ceva's theorem. Indeed, $\displaystyle\frac{a}{x}=\frac{DB}{DC},$ $\displaystyle\frac{b}{y}=\frac{EC}{EA},$ and $\displaystyle\frac{c}{z}=\frac{FA}{FB}.$ But, according to Ceva's theorem, $\displaystyle\frac{DB}{DC}\cdot\frac{EC}{EA}\cdot\frac{FA}{FB}=1.$ So, too, $\displaystyle\frac{abc}{xyz}=1.$

Since $a+b+c+x+y+z=1,$ by the Arithmetic Mean-Geometric Mean inequality, $abc\cdot xyz\le \left(\frac{1}{6}\right)^{6}.$ Since $abc=xyz,$ both do not exceed $\left(\frac{1}{6}\right)^{3},$ implying that at least one among $a,b,c$ and at least one among $x,y,z$ does not exceed $\displaystyle\frac{1}{6}.$

Acknowledgment

The statement above has been communicated to me by Leo Giugiuc, with credits to Professor Cristinel Mortici, Valahia University. The proof is by Leo Giugiuc.

508 Resource Limit Is Reached

Resource Limit Is Reached

The website is temporarily unable to service your request as it exceeded resource limit. Please try again later.

508 Resource Limit Is Reached

Resource Limit Is Reached

The website is temporarily unable to service your request as it exceeded resource limit. Please try again later.