138. Generalized "slicing" problems (own).

by Virgil Nicula, Oct 3, 2010, 4:10 PM

PP1. Let $ABC$ be a triangle. Consider a point $P$ so that $PA=AB$ , $PB=PC$ and $m\left(\widehat {PBC}\right)=x$ . Prove that exists

the relation $\sin (B-C)=2\sin C\cos (B-2sx)$ , where $s=\left\{\begin{array}{ccccc} 1 & \iff & BC & \mathrm{doesn't\ separates} & P\ ,\ A\\\ -1 & \iff & BC & \mathrm{separates} & P\ ,\ A\end{array}\right\|$ .

PP2. Let $ABC$ be a triangle for which $B=60^{\circ}+A$ . Consider two points $N\in (AC)$ , $M\in (AB)$

so that $m\left(\widehat{ACM}\right)=\frac C4$ and $m\left(\widehat{ABN}\right)=30^{\circ}$ . Prove that $m\left(\widehat{CMN}\right)=30^{\circ}$ .

PP3. Let $ABC$ be a triangle for which $B=60^{\circ}+A$ and $A\ne 30^{\circ}$ . Consider the projection $D$ of $A$ on the sideline $BC$ and the

point $E\in AC$ for which $AE=2\cdot BD$ so that : $\left\{\begin{array}{ccc} A\in (EC) & \iff & B\in (DC)\\\ E\in (AC) & \iff & B\not\in (DC)\end{array}\right\|$ . Prove that $m\left(\widehat{BEC}\right)=30^{\circ}$ .
This post has been edited 11 times. Last edited by Virgil Nicula, Dec 1, 2015, 11:10 AM

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