E:14892: Difference between revisions
No edit summary |
No edit summary |
||
| Line 14: | Line 14: | ||
Cum <math>m\left(\sphericalangle BMA\right) = 120^\circ</math>, avem <math>m\left(\sphericalangle BMP\right) = 60^\circ</math> și <math>\left[MP\right] \equiv \left[MB\right]</math>, deci triunghiul <math>BMP</math> este echilateral. | Cum <math>m\left(\sphericalangle BMA\right) = 120^\circ</math>, avem <math>m\left(\sphericalangle BMP\right) = 60^\circ</math> și <math>\left[MP\right] \equiv \left[MB\right]</math>, deci triunghiul <math>BMP</math> este echilateral. | ||
În triunghiul <math>BPR</math> avem <math>m\left(\sphericalangle RBP\right) = a + 60^\circ</math> și <math>m\left(\sphericalangle BPR\right) = 60^\circ - 2a</math>, deci <math>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)</math>. Cum <math>\sphericalangle RBP \equiv \sphericalangle PBR</math>, rezultă că triunghiul <math>PBR</math> este isoscel, cu <math>\left[ BP \right] \equiv \left[RP\right]</math>. | În triunghiul <math>BPR</math> avem <math>m\left(\sphericalangle RBP\right) = a + 60^\circ</math> și <math>m\left(\sphericalangle BPR\right) = 60^\circ - 2a</math>, deci <math>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)</math>. Cum <math>\sphericalangle RBP \equiv \sphericalangle PBR</math>, rezultă că triunghiul <math>PBR</math> este isoscel, cu <math display="block" id="eq1">\left[ BP \right] \equiv \left[RP\right].</math>Fie <math>E</math> simetricul punctului <math>M</math> față de punctul <math>P</math>. Atunci triunghiul <math>MBE</math> este dreptunghic, cu <math>m\left(\sphericalangle MBE\right) = 90^\circ</math> și <math>m\left(\sphericalangle BMP\right) = 60^\circ</math>, deci <math>m\left(\sphericalangle BEM\right) = 30^\circ = m\left(\sphericalangle BCM\right)</math>, deci patrulaterul <math>BMCE</math> este inscriptibil. | ||
Notăm <math>x= m\left(\sphericalangle CBP\right) = m\left(\sphericalangle BCP\right)</math>. Avem <math>m\left(\sphericalangle MPC\right) = m \left(\stackrel{\frown}{MC}\right) = 2\cdot m\left(\sphericalangle MBC\right) = 2\left(60^\circ - x\right)</math>. Atunci <math>m\left(\sphericalangle TPC\right) = m\left(\sphericalangle MPC\right) - m\left(\sphericalangle MPT\right) = 2\left(60^\circ - x\right) - 2b</math>. | |||
În triunghiul <math>TPC</math> avem <math>m\left(\sphericalangle TCP\right) = b + 30^\circ + x</math> și <math>m\left(\sphericalangle TPC\right) = 120^\circ - 2b - 2x</math>, deci <math>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)</math>. Cum <math>\sphericalangle TCP \equiv \sphericalangle PCT</math>, rezultă că triunghiul <math>PCT</math> este isoscel, cu <math display="block" id="eq2">\left[ CP \right] \equiv \left[TP\right].</math>Deci punctele <math>M</math>, <math>R</math>, <math>B</math>, <math>C</math>, <math>T</math> sunt conciclice. | |||
a) Avem <math>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)</math>, deci <math>\frac{1}{2} \cdot m\left(\sphericalangle RPT\right) = m\left(\sphericalangle MRT\right) + m\left(\sphericalangle MTR\right).</math> | |||
b) Avem <math>m\left(\sphericalangle ARM\right) = \frac{1}{2}\cdot m\left(\stackrel{\frown}{BM}\right) = m\left(\sphericalangle BCM\right) = 30^\circ.</math> | |||
Revision as of 20:18, 20 December 2023
E:14892 (Radu Pop & Ienuțaș Vasile)
Fie triunghiul 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 C\right) > 30^\circ} și 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 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 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 MPR\right) = 2a} 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 MPT\right) = 2b} .
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 m\left(\sphericalangle BMA\right) = 120^\circ} , 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 BMP\right) = 60^\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 \left[MP\right] \equiv \left[MB\right]} , deci 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 BMP} este echilateral.
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 BPR} 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 RBP\right) = a + 60^\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 BPR\right) = 60^\circ - 2a} , 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 BRP\right) = 180^\circ - \left(60^\circ + a\right) - \left(60^\circ -2a\right) = 60^\circ + a = m\left(\sphericalangle RBP\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 RBP \equiv \sphericalangle PBR} , 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 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(\stackrel{\frown}{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(\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.}