E:14892

E:14892 (Radu Pop & Ienuțaș Vasile)

Fie triunghiul $ABC$ cu $m\left(\sphericalangle C\right)>30^{\circ }$ și punctele $M$ , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . Punctul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} este situat în interiorul triunghiului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ABC} astfel încât Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle BMA\right) = 120^\circ} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle BCM\right) = 30^\circ} , punctul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P\in \left(MD\right.} astfel încât Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[MP\right] \equiv \left[MB\right]} cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AM \cap BC = \left\{D\right\}} , iar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R\in \left(AB\right)} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \in \left(AC\right)} astfel încât Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle RBM\right) = \frac{1}{2} \cdot m\left(\sphericalangle RPM\right)} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle TPM\right) = 2 \cdot m\left(\sphericalangle TCM\right)} .

Arătați că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} \cdot m\left(\sphericalangle RPT\right) = m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MTR\right)}
Determinați măsura unghiului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sphericalangle ARM}
Arătați că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MAT\right) = m\left(\sphericalangle DMC\right)}

Soluție miniatura

Folosim notațiile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle RBM\right) = a} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle TCM\right) = b} . Atunci $m\left(\sphericalangle MPR\right)=2a$ și $m\left(\sphericalangle MPT\right)=2b$ .

Cum $m\left(\sphericalangle BMA\right)=120^{\circ }$ , avem $m\left(\sphericalangle BMP\right)=60^{\circ }$ și $\left[MP\right]\equiv \left[MB\right]$ , deci triunghiul $BMP$ este echilateral.

În triunghiul $BPR$ avem $m\left(\sphericalangle RBP\right)=a+60^{\circ }$ și $m\left(\sphericalangle BPR\right)=60^{\circ }-2a$ , deci $m\left(\sphericalangle BRP\right)=180^{\circ }-\left(60^{\circ }+a\right)-\left(60^{\circ }-2a\right)=60^{\circ }+a=m\left(\sphericalangle RBP\right)$ . Cum $\sphericalangle RBP\equiv \sphericalangle PBR$ , rezultă că triunghiul $PBR$ este isoscel, cu

\left[BP\right]\equiv \left[RP\right].

Fie

E

simetricul punctului

M

față de punctul

P

. Atunci triunghiul

MBE

este dreptunghic, cu

m\left(\sphericalangle MBE\right)=90^{\circ }

și

m\left(\sphericalangle BMP\right)=60^{\circ }

, deci

m\left(\sphericalangle BEM\right)=30^{\circ }=m\left(\sphericalangle BCM\right)

, deci patrulaterul

BMCE

este inscriptibil.

Notăm $x=m\left(\sphericalangle CBP\right)=m\left(\sphericalangle BCP\right)$ . Avem $m\left(\sphericalangle MPC\right)=m\left({\stackrel {\frown }{MC}}\right)=2\cdot m\left(\sphericalangle MBC\right)=2\left(60^{\circ }-x\right)$ . Atunci $m\left(\sphericalangle TPC\right)=m\left(\sphericalangle MPC\right)-m\left(\sphericalangle MPT\right)=2\left(60^{\circ }-x\right)-2b$ .

În triunghiul $TPC$ avem $m\left(\sphericalangle TCP\right)=b+30^{\circ }+x$ și $m\left(\sphericalangle TPC\right)=120^{\circ }-2b-2x$ , deci $m\left(\sphericalangle PTC\right)=180^{\circ }-\left(b+30^{\circ }+x\right)-\left(120^{\circ }-2b-2x\right)=30^{\circ }+b+x=m\left(\sphericalangle TCP\right)$ . Cum $\sphericalangle TCP\equiv \sphericalangle PCT$ , rezultă că triunghiul $PCT$ este isoscel, cu

\left[CP\right]\equiv \left[TP\right].

Deci punctele

M

,

R

, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} sunt conciclice.

a) Avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle RPT\right) = m \left(\stackrel{\frown}{RT}\right) = m\left(\stackrel{\frown}{RM}\right) + m\left(\stackrel{\frown}{MT}\right) = 2\cdot m\left(\sphericalangle MTR\right) + 2\cdot m\left(\sphericalangle MRT\right)} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} \cdot m\left(\sphericalangle RPT\right) = m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MTR\right).}

b) Avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\left(\sphericalangle ARM\right) = \frac{1}{2}\cdot m\left(\stackrel{\frown}{BM}\right) = m\left(\sphericalangle BCM\right) = 30^\circ.}