28354: Difference between revisions
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''Fie <math>O</math> punctul de intersecție a diagonalelor patrulaterului convex <math>ABCD</math> și punctele <math>E</math>, <math>F</math>, <math>G</math> și <math>H</math> situate pe segmentele <math>OA</math>, <math>OB</math>, <math>OC</math>, respectiv <math>OD</math>, astfel încât <math>AE = BF = CG = DH</math>. Notăm cu <math>I</math>, <math>J</math>, <math>K</math> și <math>L</math> mijloacele segmentelor <math>AB</math>, <math>BC</math>, <math>CD</math>, respectiv <math>DA</math> și cu <math>M</math>,<math>N</math>, <math>P</math> și <math>Q</math> mijloacele segmentelor <math>EF</math>, <math>FG</math>, | ''Fie <math>O</math> punctul de intersecție a diagonalelor patrulaterului convex <math>ABCD</math> și punctele <math>E</math>, <math>F</math>, <math>G</math> și <math>H</math> situate pe segmentele <math>OA</math>, <math>OB</math>, <math>OC</math>, respectiv <math>OD</math>, astfel încât <math>AE = BF = CG = DH</math>. Notăm cu <math>I</math>, <math>J</math>, <math>K</math> și <math>L</math> mijloacele segmentelor <math>AB</math>, <math>BC</math>, <math>CD</math>, respectiv <math>DA</math> și cu <math>M</math>,<math>N</math>, <math>P</math> și <math>Q</math> mijloacele segmentelor <math>EF</math>, <math>FG</math>, | ||
<math>GH</math>, respectiv <math>HE</math>. Arătați că: | <math>GH</math>, respectiv <math>HE</math>. Arătați că: | ||
<li> a) punctele <math>I</math>,<math>M</math> și <math>K</math> sunt coliniare dacă și numai dacă <math>AC=BD</math>. | <ol><li> a) punctele <math>I</math>,<math>M</math> și <math>K</math> sunt coliniare dacă și numai dacă <math>AC=BD</math>.</li> | ||
<li> b) <math>AC \not= BD</math>, punctele de intersecție ale dreptelor <math>IM</math>,<math>NJ</math>,<math>PK</math> și <math>LQ</math> sunt vârfurile unui dreptunghi.</li></ol>'' | <li> b) <math>AC \not= BD</math>, punctele de intersecție ale dreptelor <math>IM</math>,<math>NJ</math>,<math>PK</math> și <math>LQ</math> sunt vârfurile unui dreptunghi.</li></ol>'' | ||
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\overrightarrow{i} + \frac{{BD}}{2} \cdot \overrightarrow{j} </math> (2) | \overrightarrow{i} + \frac{{BD}}{2} \cdot \overrightarrow{j} </math> (2) | ||
Din (1) și (2) rezultă ca <math>I</math>,<math>M</math> și <math>K</math> sunt coliniare dacă și numai dacă <math>AC = BD</math>. | Din (1) și (2) rezultă ca <math>I</math>, <math>M</math> și <math>K</math> sunt coliniare dacă și numai dacă <math>AC = BD</math>. | ||
b) Notăm <math>\overrightarrow{i} +\overrightarrow{j} = \overrightarrow{O R}</math> și <math>\overrightarrow{-i} +\overrightarrow{j} = \overrightarrow{O S}</math>. | b) Notăm <math>\overrightarrow{i} +\overrightarrow{j} = \overrightarrow{O R}</math> și <math>\overrightarrow{-i} +\overrightarrow{j} = \overrightarrow{O S}</math>. | ||
Se observă că semidreptele <math>(OR</math> și <math>(OS</math> sunt bisectoarele unghiurilor <math>COD</math>, respectiv <math>AOD</math>. Ca în (1),deducem că <math>\overrightarrow{P K} =\overrightarrow{I M} = \frac{{x}}{2} \cdot (\overrightarrow{i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O R}</math>,iar <math>\overrightarrow{J N} =\overrightarrow{Q L} = \frac{{x}}{2} \cdot (\overrightarrow{-i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O S}</math>. | Se observă că semidreptele <math>(OR</math> și <math>(OS</math> sunt bisectoarele unghiurilor <math>COD</math>, respectiv <math>AOD</math>. Ca în (1),deducem că <math>\overrightarrow{P K} =\overrightarrow{I M} = \frac{{x}}{2} \cdot (\overrightarrow{i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O R}</math>, iar <math>\overrightarrow{J N} =\overrightarrow{Q L} = \frac{{x}}{2} \cdot (\overrightarrow{-i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O S}</math>. | ||
Fiind bisectoarele a două unghiuri adiacente suplementare, semidreptele (OR și OS sunt perpendiculare ,de unde rezultă că <math>IM \perp JN</math>,<math>JN \perp KP</math>, <math>KP \perp LQ</math> și <math>LQ \perp IM</math>.Dar <math>AC \not= BD</math> , deci <math>I</math>, <math>M</math> și <math>K</math> sunt necoliniare, așadar <math>IM \parallel KP</math>, și analog <math>JN \parallel LQ</math>. Notând cu <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> intersecțiile perechilor de drepte <math>IM</math> și <math>JN</math>, <math>JN</math> și <math>KP</math>, <math>KP</math> și <math>LQ</math>, <math>LQ</math> și <math>IM</math>, din cele de mai înaite rezultă că <math>XYZW</math> este dreptunghi. | Fiind bisectoarele a două unghiuri adiacente suplementare, semidreptele (OR și OS sunt perpendiculare, de unde rezultă că <math>IM \perp JN</math>,<math>JN \perp KP</math>, <math>KP \perp LQ</math> și <math>LQ \perp IM</math>. Dar <math>AC \not= BD</math>, deci <math>I</math>, <math>M</math> și <math>K</math> sunt necoliniare, așadar <math>IM \parallel KP</math>, și analog <math>JN \parallel LQ</math>. Notând cu <math>X</math>, <math>Y</math>, <math>Z</math>, <math>W</math> intersecțiile perechilor de drepte <math>IM</math> și <math>JN</math>, <math>JN</math> și <math>KP</math>, <math>KP</math> și <math>LQ</math>, <math>LQ</math> și <math>IM</math>, din cele de mai înaite rezultă că <math>XYZW</math> este dreptunghi. | ||
Revision as of 13:28, 4 December 2023
28354 (Florin Bojor)
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 O} punctul de intersecție a diagonalelor patrulaterului convex 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 ABCD} 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 E} , 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 F} , 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 G} 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 H} situate pe segmentele 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 OA} , 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 OB} , 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 OC} , 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 OD} , 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 AE = BF = CG = DH} . Notăm 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 I} , 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 J} , 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 K} 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 L} mijloacele segmentelor 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} , 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} , 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 CD} , 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 DA} și 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} ,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} , 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} 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 Q} mijloacele segmentelor 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 EF} , 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 FG} , 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 GH} , 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 HE} . Arătați că:
- a) 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 I} ,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 K} sunt coliniare dacă și numai dacă 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=BD} .
- 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 AC \not= BD} , punctele de intersecție ale dreptelor 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 IM} ,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 NJ} ,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 PK} 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 LQ} sunt vârfurile unui dreptunghi.
Soluție.
a)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 AE = BF = CG = x}
și versorii 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 \overrightarrow{i}}
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 \overrightarrow{j}}
ai vectorilor 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 \overrightarrow{A C}}
, 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 \overrightarrow{B D}}
.
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 I} 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} sunt mijloacele segmentelor 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} , 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 EF} , obținem:
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 \overrightarrow{I M} = \frac{{1}}{2} \cdot (\overrightarrow{A E}+\overrightarrow{B F}) =\frac{{x}}{2} \cdot \overrightarrow{i} + \frac{{x}}{2} \cdot \overrightarrow{j} } . (1)
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 K} este mijloxul segemntului 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 CD} ,deducem:
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 \overrightarrow{I K} = \frac{{1}}{2} \cdot (\overrightarrow{A C}+\overrightarrow{B D}) =\frac{{AC}}{2} \cdot \overrightarrow{i} + \frac{{BD}}{2} \cdot \overrightarrow{j} }
(2)
Din (1) și (2) rezultă ca 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 I} , 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 K} sunt coliniare dacă și numai dacă 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 = BD} .
b) 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 \overrightarrow{i} +\overrightarrow{j} = \overrightarrow{O R}} 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 \overrightarrow{-i} +\overrightarrow{j} = \overrightarrow{O S}} .
Se observă că semidreptele 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 (OR} 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 (OS} sunt bisectoarele unghiurilor 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 COD} , 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 AOD} . Ca în (1),deducem 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 \overrightarrow{P K} =\overrightarrow{I M} = \frac{{x}}{2} \cdot (\overrightarrow{i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O R}} , 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 \overrightarrow{J N} =\overrightarrow{Q L} = \frac{{x}}{2} \cdot (\overrightarrow{-i}+\overrightarrow{j}) =\frac{{x}}{2} \cdot \overrightarrow{O S}} .
Fiind bisectoarele a două unghiuri adiacente suplementare, semidreptele (OR și OS sunt perpendiculare, de unde 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 IM \perp JN} ,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 JN \perp KP} , 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 KP \perp LQ} 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 LQ \perp IM} . Dar 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 \not= BD} , 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 I} , 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 K} sunt necoliniare, 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 IM \parallel KP} , și analog 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 JN \parallel LQ} . Notând 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 X} , 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 Y} , 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 Z} , 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 W} intersecțiile perechilor de drepte 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 IM} 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 JN} , 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 JN} 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 KP} , 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 KP} 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 LQ} , 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 LQ} 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 IM} , din cele de mai înaite 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 XYZW} este dreptunghi.