E:16203: Difference between revisions

Revision as of 10:36, 29 February 2024

E:16203 (Dana Heuberger)

Fie 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 BCD} dreptunghic în 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 D} , 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 \sphericalangle CBD = 90^\circ} . Se consideră 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} astfel încât semidreapta $CD$ este bisectoarea $\sphericalangle BCM$ și $MD\bot BC$ . Fie punctul $L$ astfel încât $B$ se află pe segmentul $ML$ și $BM=2BL$ . Notăm cu $F$ simetricul lui $D$ față de $B$ . Arătați că

a) $MB=CF$

b) $\sphericalangle BDL=\sphericalangle BMD$

Soluție:

Fie $BD\cap CM=\left\{A\right\}$ . Atunci triunghiul $ABC$ este echilateral. Notăm $AB=a>0$ . Deoarece $CD$ este înălțime a triunghiului echilateral $ABC$ , rezultă că $CD$ este și bisectoare a $\sphericalangle ACB$ .

Fie $BC\cap DM=\left\{E\right\}$ . Se arată ușor că $BE={\frac {a}{4}}$ , 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 EC= \frac{3a}{4}} . Din triunghiul dreptunghic 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 CEM} rezultă 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 EC = \frac{MC}{2}} , așadar 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 CM= \frac{3a}{2}} .

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 MA=MC-AC=\frac{a}{2}=BF} , 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 AB= BC =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 \sphericalangle MAB = \sphericalangle FBC = 120^\circ} , deci triunghiurile 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 ABM} 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 BCF} sunt congruente, așadar 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 MB=CF} .

b) Triunghiurile 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 ABM} 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 BCF} sunt congruente, de unde obținem 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 \sphericalangle MBA = \sphericalangle FCB = x^\circ} . Rezultă 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 \sphericalangle MBC = \sphericalangle ACF = 60^\circ + x^\circ} .

Deoarece 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{MB}{BL} = \frac{AB}{BF} = 2 } , 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 \sphericalangle MBA = \sphericalangle LBF} , rezulră că triunghiurile 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 MBA} 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 LBF} sunt asemenea, 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 FL \parallel MC} . Folosind secanta 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 FC} , deducem că ungiurile alterne interne 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 CFL} 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 ACF} sunt congruente, așadar 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 ACF = \sphericalangle MBC = \sphericalangle CFL} . Din 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 \Delta MBA \approx \Delta LBF} rezultă 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 FL = \frac{MA}{2} = \frac{a}{4}=BE} .

Revision as of 10:31, 29 February 2024 view source Andrei.Horvat (talk \| contribs) 700 edits No edit summary Tag: Visual edit ← Older edit		Revision as of 10:36, 29 February 2024 view source Andrei.Horvat (talk \| contribs) 700 edits No edit summary Tag: Visual edit Newer edit →
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	b) Triunghiurile <math>ABM</math> și <math>BCF</math> sunt congruente, de unde obținem că <math>\sphericalangle MBA = \sphericalangle FCB = x^\circ</math>. Rezultă că <math>\sphericalangle MBC = \sphericalangle ACF = 60^\circ + x^\circ</math>.		b) Triunghiurile <math>ABM</math> și <math>BCF</math> sunt congruente, de unde obținem că <math>\sphericalangle MBA = \sphericalangle FCB = x^\circ</math>. Rezultă că <math>\sphericalangle MBC = \sphericalangle ACF = 60^\circ + x^\circ</math>.

	Deoarece <math>\frac{MB}{BL} = \frac{AB}{BF} = 2 </math>, iar <math>\sphericalangle MBA = \sphericalangle LBF</math>, rezulră că triunghiurile <math>MBA</math> și <math>LBF</math> sunt asemenea, deci <math>FL \parallel MC</math>. Folosind secanta		Deoarece <math>\frac{MB}{BL} = \frac{AB}{BF} = 2 </math>, iar <math>\sphericalangle MBA = \sphericalangle LBF</math>, rezulră că triunghiurile <math>MBA</math> și <math>LBF</math> sunt asemenea, deci <math>FL \parallel MC</math>. Folosind secanta <math>FC</math>, deducem că ungiurile alterne interne <math>CFL</math> și <math>ACF</math> sunt congruente, așadar <math>\sphericalangle ACF = \sphericalangle MBC = \sphericalangle CFL</math>. Din <math>\Delta MBA \approx \Delta LBF</math> rezultă că <math>FL = \frac{MA}{2} = \frac{a}{4}=BE</math>.