E:14892

E:14892 (Radu Pop & Ienuțaș Vasile)

Fie triunghiul $ABC$ cu $m\left(\sphericalangle C\right)>30^{\circ }$ și punctele $M$, $P$, $R$, $T$. Punctul $M$ este situat în interiorul triunghiului $ABC$ astfel încât $m\left(\sphericalangle BMA\right)=120^{\circ }$ și $m\left(\sphericalangle BCM\right)=30^{\circ }$, punctul $P\in \left(MD\right.$ astfel încât $\left[MP\right]\equiv \left[MB\right]$ cu $AM\cap BC=\left\{D\right\}$, iar $R\in \left(AB\right)$ și $T\in \left(AC\right)$ astfel încât $m\left(\sphericalangle RBM\right)={\frac {1}{2}}\cdot m\left(\sphericalangle RPM\right)$ și $m\left(\sphericalangle TPM\right)=2\cdot m\left(\sphericalangle TCM\right)$.

1. Arătați că ${\frac {1}{2}}\cdot m\left(\sphericalangle RPT\right)=m\left(\sphericalangle MRT\right)+m\left(\sphericalangle MTR\right)$
2. Determinați măsura unghiului $\sphericalangle ARM$
3. Arătați că $m\left(\sphericalangle MRT\right)+m\left(\sphericalangle MAT\right)=m\left(\sphericalangle DMC\right)$

Soluție miniatura

Folosim notațiile $m\left(\sphericalangle RBM\right)=a$ și $m\left(\sphericalangle TCM\right)=b$. 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 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[ BP \right] \equiv \left[RP\right]} .

Fie 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 E} simetricul punctului 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} față de 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} . Atunci triunghiul 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 MBE} este dreptunghic, 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 m\left(\sphericalangle MBE\right) = 90^\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 BMP\right) = 60^\circ} , 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 m\left(\sphericalangle BEM\right) = 30^\circ = m\left(\sphericalangle BCM\right)} , deci patrulaterul 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 BMCE} este inscriptibil.

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

În triunghiul 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 TPC} 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 TCP\right) = b + 30^\circ + x} 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 TPC\right) = 120^\circ - 2b - 2x} , 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 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 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 TCP \equiv \sphericalangle PCT} , rezultă că triunghiul 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 PCT} este isoscel, 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 \left[ CP \right] \equiv \left[TP\right]} .

Deci punctele 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} , 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 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(\widearc{RT}\right) = m\left(\widearc{RM}\right) + m\left(\widearc{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(\widearc{BM}\right) = m\left(\sphericalangle BCM\right) = 30^\circ.}