E:16899: Difference between revisions

Revision as of 15:13, 19 August 2025

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 $\left(AC\right)$ astfel încât $BM=CN$ . Fie punctul $P$ astfel încât $NM=MP$ , 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 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 PMB \equiv \Delta MNC} , 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 \sphericalangle PBC = \sphericalangle MCN} .

Cum 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 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 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.} .

@@ Line 15: / Line 15: @@
 \end{cases}</math>, conform cazului de congruenţă L.U.L., rezultă <math>\Delta PMB \equiv \Delta MNC</math>, deci <math>\sphericalangle PBC = \sphericalangle MCN</math>.
-Cum punctele <math>B</math>, <math>M</math> și <math>C</math>  sunt colinare, avem
+Cum punctele <math>B</math>, <math>M</math> și <math>C</math>  sunt colinare, avem <math display="block" id="e16899-1" qid="e16899-1">\sphericalangle BMP + \sphericalangle PMN + \sphericalangle NMC = 180^\circ.
-<math>\sphericalangle BMP + \sphericalangle PMN + \sphericalangle NMC = 180^\circ.
+</math>
-</math>  în triunghiul <math>\Delta CMN</math> avem <math>\sphericalangle CNM + \sphericalangle NMC + \sphericalangle MCN = 180^\circ.
-</math> Din \eqref{ang1} și \eqref{ang2} și <math>\sphericalangle PMB = \sphericalangle MNC</math> se deduce <math>\sphericalangle PMN = \sphericalangle MCN</math>.
-în concluzie, <math>\sphericalangle NMP = \sphericalangle PBM = \sphericalangle MCA</math>.
+În triunghiul <math>\Delta CMN</math> avem <math display="block" id="e16899-2" qid="e16899-2">\sphericalangle CNM + \sphericalangle NMC + \sphericalangle MCN = 180^\circ.
+</math>
+Din cele două egalități și <math>\sphericalangle PMB = \sphericalangle MNC</math> se deduce <math>\sphericalangle PMN = \sphericalangle MCN</math>.
+În concluzie, are loc egalitatea <math display="block">\sphericalangle NMP = \sphericalangle PBM = \sphericalangle MCA.</math> .