E:16899: Difference between revisions
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\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>. | \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 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> . | |||
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 este mai mare decât lungimea segmentului . 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 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 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} 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 P} sunt de aceeași parte a dreptei 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} , iar distanţa de la 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} la 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 aceeași cu distanţa de la 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} la 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 AC} . Arătaţi 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 NMP = \sphericalangle PBM = \sphericalangle MCA} .
![{\displaystyle \left[AC\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75221e2f8dfb66a48aec1498c809c414321762ff)
Soluție
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 \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 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 PP_1M \equiv \Delta MM_1N}
, 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 PMB = \sphericalangle MNC}
.
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 \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.} .