E:16899

E:16899 (Angela Lopată)

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 ABC} un triunghi pentru care lungimea proiecţiei laturii 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} pe dreapta 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 BC} este mai mare decât lungimea segmentului 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[AC\right]} . Considerăm 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 N} pe laturile 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(BC\right)} , respectiv 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(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 BM = CN} . Fie 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} 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 NM = MP} , punctele $N$ și $P$ sunt de aceeași parte a dreptei $BC$, iar distanţa de la punctul $P$ la dreapta $BC$ este aceeași cu distanţa de la punctul $M$ la dreapta $AC$. Arătaţi că $\sphericalangle NMP=\sphericalangle PBM=\sphericalangle MCA$.

Soluție.

Din ${\begin{cases}PM=MN\\PP_{1}=MM_{1}\\\sphericalangle P_{1}=\sphericalangle M_{1}=90^{\circ }\end{cases}}$, conform cazului de congruenţă C.I., rezultă $\Delta PP_{1}M\equiv \Delta MM_{1}N$, deci $\sphericalangle PMB=\sphericalangle MNC$.

Din ${\begin{cases}PM=MN\\MB=NC\\\sphericalangle PMB=\sphericalangle MNC\end{cases}}$, conform cazului de congruenţă L.U.L., rezultă $\Delta PMB\equiv \Delta MNC$, deci $\sphericalangle PBC=\sphericalangle MCN$.

Cum punctele $B$, $M$ 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 C} sunt colinare, 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 \sphericalangle BMP + \sphericalangle PMN + \sphericalangle NMC = 180^\circ. }

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 \Delta CMN} 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 \sphericalangle CNM + \sphericalangle NMC + \sphericalangle MCN = 180^\circ. }

Din cele două egalități 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 PMB = \sphericalangle MNC} se deduce 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 PMN = \sphericalangle MCN} .

În concluzie, are loc egalitatea 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 NMP = \sphericalangle PBM = \sphericalangle MCA.}