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196) + defining Unicode char U+00C5 (decimal 197) + defining Unicode char U+00C6 (decimal 198) + defining Unicode char U+00C7 (decimal 199) + defining Unicode char U+00C8 (decimal 200) + defining Unicode char U+00C9 (decimal 201) + defining Unicode char U+00CA (decimal 202) + defining Unicode char U+00CB (decimal 203) + defining Unicode char U+00CC (decimal 204) + defining Unicode char U+00CD (decimal 205) + defining Unicode char U+00CE (decimal 206) + defining Unicode char U+00CF (decimal 207) + defining Unicode char U+00D0 (decimal 208) + defining Unicode char U+00D1 (decimal 209) + defining Unicode char U+00D2 (decimal 210) + defining Unicode char U+00D3 (decimal 211) + defining Unicode char U+00D4 (decimal 212) + defining Unicode char U+00D5 (decimal 213) + defining Unicode char U+00D6 (decimal 214) + defining Unicode char U+00D8 (decimal 216) + defining Unicode char U+00D9 (decimal 217) + defining Unicode char U+00DA (decimal 218) + defining Unicode char U+00DB (decimal 219) + defining Unicode char U+00DC (decimal 220) + defining Unicode char U+00DD (decimal 221) + defining Unicode char U+00DE (decimal 222) + defining Unicode char U+00DF (decimal 223) + defining Unicode char U+00E0 (decimal 224) + defining Unicode char U+00E1 (decimal 225) + defining Unicode char U+00E2 (decimal 226) + defining Unicode char U+00E3 (decimal 227) + defining Unicode char U+00E4 (decimal 228) + defining Unicode char U+00E5 (decimal 229) + defining Unicode char U+00E6 (decimal 230) + defining Unicode char U+00E7 (decimal 231) + defining Unicode char U+00E8 (decimal 232) + defining Unicode char U+00E9 (decimal 233) + defining Unicode char U+00EA (decimal 234) + defining Unicode char U+00EB (decimal 235) + defining Unicode char U+00EC (decimal 236) + defining Unicode char U+00ED (decimal 237) + defining Unicode char U+00EE (decimal 238) + defining Unicode char U+00EF (decimal 239) + defining Unicode char U+00F0 (decimal 240) + defining Unicode char U+00F1 (decimal 241) + defining Unicode char U+00F2 (decimal 242) + defining Unicode char U+00F3 (decimal 243) + defining Unicode char U+00F4 (decimal 244) + defining Unicode char U+00F5 (decimal 245) + defining Unicode char U+00F6 (decimal 246) + defining Unicode char U+00F8 (decimal 248) + defining Unicode char U+00F9 (decimal 249) + defining Unicode char U+00FA (decimal 250) + defining Unicode char U+00FB (decimal 251) + defining Unicode char U+00FC (decimal 252) + defining Unicode char U+00FD (decimal 253) + defining Unicode char U+00FE (decimal 254) + defining Unicode char U+00FF (decimal 255) + defining Unicode char U+0100 (decimal 256) + defining Unicode char U+0101 (decimal 257) + defining Unicode char U+0102 (decimal 258) + defining Unicode char U+0103 (decimal 259) + defining Unicode char U+0104 (decimal 260) + defining Unicode char U+0105 (decimal 261) + defining Unicode char U+0106 (decimal 262) + defining Unicode char U+0107 (decimal 263) + defining Unicode char U+0108 (decimal 264) + defining Unicode char U+0109 (decimal 265) + defining Unicode char U+010A (decimal 266) + defining Unicode char U+010B (decimal 267) + defining Unicode char U+010C (decimal 268) + defining Unicode char U+010D (decimal 269) + defining Unicode char U+010E (decimal 270) + defining Unicode char U+010F (decimal 271) + defining Unicode char U+0110 (decimal 272) + defining Unicode char U+0111 (decimal 273) + defining Unicode char U+0112 (decimal 274) + defining Unicode char U+0113 (decimal 275) + defining Unicode char U+0114 (decimal 276) + defining Unicode char U+0115 (decimal 277) + defining Unicode char U+0116 (decimal 278) + defining Unicode char U+0117 (decimal 279) + defining Unicode char U+0118 (decimal 280) + defining Unicode char U+0119 (decimal 281) + defining Unicode char U+011A (decimal 282) + defining Unicode char U+011B (decimal 283) + defining Unicode char U+011C (decimal 284) + defining Unicode char U+011D (decimal 285) + defining Unicode char U+011E (decimal 286) + defining Unicode char U+011F (decimal 287) + defining Unicode char U+0120 (decimal 288) + defining Unicode char U+0121 (decimal 289) + defining Unicode char U+0122 (decimal 290) + defining Unicode char U+0123 (decimal 291) + defining Unicode char U+0124 (decimal 292) + defining Unicode char U+0125 (decimal 293) + defining Unicode char U+0128 (decimal 296) + defining Unicode char U+0129 (decimal 297) + defining Unicode char U+012A (decimal 298) + defining Unicode char U+012B (decimal 299) + defining Unicode char U+012C (decimal 300) + defining Unicode char U+012D (decimal 301) + defining Unicode char U+012E (decimal 302) + defining Unicode char U+012F (decimal 303) + defining Unicode char U+0130 (decimal 304) + defining Unicode char U+0131 (decimal 305) + defining Unicode char U+0132 (decimal 306) + defining Unicode char U+0133 (decimal 307) + defining Unicode char U+0134 (decimal 308) + defining Unicode char U+0135 (decimal 309) + defining Unicode char U+0136 (decimal 310) + defining Unicode char U+0137 (decimal 311) + defining Unicode char U+0139 (decimal 313) + defining Unicode char U+013A (decimal 314) + defining Unicode char U+013B (decimal 315) + defining Unicode char U+013C (decimal 316) + defining Unicode char U+013D (decimal 317) + defining Unicode char U+013E (decimal 318) + defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0145 (decimal 325) + defining Unicode char U+0146 (decimal 326) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+014C (decimal 332) + defining Unicode char U+014D (decimal 333) + defining Unicode char U+014E (decimal 334) + defining Unicode char U+014F (decimal 335) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0156 (decimal 342) + defining Unicode char U+0157 (decimal 343) + defining Unicode char U+0158 (decimal 344) + defining Unicode char U+0159 (decimal 345) + defining Unicode char U+015A (decimal 346) + defining Unicode char U+015B (decimal 347) + defining Unicode char U+015C (decimal 348) + defining Unicode char U+015D (decimal 349) + defining Unicode char U+015E (decimal 350) + defining Unicode char U+015F (decimal 351) + defining Unicode char U+0160 (decimal 352) + defining Unicode char U+0161 (decimal 353) + defining Unicode char U+0162 (decimal 354) + defining Unicode char U+0163 (decimal 355) + defining Unicode char U+0164 (decimal 356) + defining Unicode char U+0165 (decimal 357) + defining Unicode char U+0168 (decimal 360) + defining Unicode char U+0169 (decimal 361) + defining Unicode char U+016A (decimal 362) + defining Unicode char U+016B (decimal 363) + defining Unicode char U+016C (decimal 364) + defining Unicode char U+016D (decimal 365) + defining Unicode char U+016E (decimal 366) + defining Unicode char U+016F (decimal 367) + defining Unicode char U+0170 (decimal 368) + defining Unicode char U+0171 (decimal 369) + defining Unicode char U+0172 (decimal 370) + defining Unicode char U+0173 (decimal 371) + defining Unicode char U+0174 (decimal 372) + defining Unicode char U+0175 (decimal 373) + defining Unicode char U+0176 (decimal 374) + defining Unicode char U+0177 (decimal 375) + defining Unicode char U+0178 (decimal 376) + defining Unicode char U+0179 (decimal 377) + defining Unicode char U+017A (decimal 378) + defining Unicode char U+017B (decimal 379) + defining Unicode char U+017C (decimal 380) + defining Unicode char U+017D (decimal 381) + defining Unicode char U+017E (decimal 382) + defining Unicode char U+01CD (decimal 461) + defining Unicode char U+01CE (decimal 462) + defining Unicode char U+01CF (decimal 463) + defining Unicode char U+01D0 (decimal 464) + defining Unicode char U+01D1 (decimal 465) + defining Unicode char U+01D2 (decimal 466) + defining Unicode char U+01D3 (decimal 467) + defining Unicode char U+01D4 (decimal 468) + defining Unicode char U+01E2 (decimal 482) + defining Unicode char U+01E3 (decimal 483) + defining Unicode char U+01E6 (decimal 486) + defining Unicode char U+01E7 (decimal 487) + defining Unicode char U+01E8 (decimal 488) + defining Unicode char U+01E9 (decimal 489) + defining Unicode char U+01EA (decimal 490) + defining Unicode char U+01EB (decimal 491) + defining Unicode char U+01F0 (decimal 496) + defining Unicode char U+01F4 (decimal 500) + defining Unicode char U+01F5 (decimal 501) + defining Unicode char U+0218 (decimal 536) + defining Unicode char U+0219 (decimal 537) + defining Unicode char U+021A (decimal 538) + defining Unicode char U+021B (decimal 539) + defining Unicode char U+1E02 (decimal 7682) + defining Unicode char U+1E03 (decimal 7683) + defining Unicode char U+200C (decimal 8204) + defining Unicode char U+2010 (decimal 8208) + defining Unicode char U+2011 (decimal 8209) + defining Unicode char U+2012 (decimal 8210) + defining Unicode char U+2013 (decimal 8211) + defining Unicode char U+2014 (decimal 8212) + defining Unicode char U+2015 (decimal 8213) + defining Unicode char U+2018 (decimal 8216) + defining Unicode char U+2019 (decimal 8217) + defining Unicode char U+201A (decimal 8218) + defining Unicode char U+201C (decimal 8220) + defining Unicode char U+201D (decimal 8221) + defining Unicode char U+201E (decimal 8222) + defining Unicode char U+2030 (decimal 8240) + defining Unicode char U+2031 (decimal 8241) + defining Unicode char U+2039 (decimal 8249) + defining Unicode char U+203A (decimal 8250) + defining Unicode char U+2423 (decimal 9251) +) +Now handling font encoding OT1 ... +... processing UTF-8 mapping file for font encoding OT1 + 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z%PWc6^IZ6`Y~@d%$czhfoZDtn3&eU(!>%lnHkaIKcKyp^g$M9r;f|U0GT5!LnCIqs zdnmq$L#IDQmKUj+Uq;ni*)msjUa$_i$9ObhWc{{\draw [help lines] (0,0) (6,6);} + \draw[thick] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \draw[thick] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \node at (3,3) {$c^2$}; + \node at (4,1.5) {c}; + \node[below] at (4,0) {a}; + \node [right] at (6,1) {b}; +\end{tikzpicture} +\end{center} +\end{frame} + +\begin{frame} +Dany jest trójkąt prostokątny o bokach długości a, b i c. Konstruujemy kwadrat o boku długości a+b w sposób ukazany po lewej stronie, a następnie z prawej. Z jednej strony pole kwadratu równe jest sumie pól czterech trójkątów prostokątnych i kwadratu zbudowanego na ich przeciwprostokątnych, z drugiej zaś równe jest ono sumie pól tych samych czterech trójkątów i dwóch mniejszych kwadratów zbudowanych na ich przyprostokątnych. Stąd wniosek, że pole kwadratu zbudowanego na przeciwprostokątnej jest równe sumie pól kwadratów zbudowanych na przyprostokątnych. +\begin{center} +\begin{tikzpicture}[scale=0.5] + \path [fill=blue] (0,0) rectangle (6,6); + \uncover<1>{\draw [help lines] (0,0) (6,6);} + \draw[thick] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \draw[thick] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \node at (3,3) {$c^2$}; + \node at (4,1.5) {c}; + \node[below] at (4,0) {a}; + \node [right] at (6,1) {b}; +\end{tikzpicture} +\begin{tikzpicture}[scale=0.5] + \path [fill=blue] (0,0) rectangle (6,6); + \uncover<1>{\draw [help lines] (0,0) (6,6);} + \draw[thick] (0,0) -- (4,2) -- (6,6); + \draw[thick] (0,2) -- (4,2) -- (4,6); + \draw[thick] (4,0) -- (4,2) -- (6,2); + \draw[thick] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (0,2) -- (4,2) -- (4,6) -- (0,6) -- (0,2); + \path [fill=white] (4,0) -- (4,2) -- (6,2) -- (6,0) -- (4,0); + \node[below] at (2,4.5) {$a^2$}; + \node [right] at (4.5,1) {$b^2$}; +\end{tikzpicture} +\end{center} + +\end{frame} + +\end{document} \ No newline at end of file diff --git a/04/tw Pitagorasa/Twierdzenie Pitagorasa.tex.bak b/04/tw Pitagorasa/Twierdzenie Pitagorasa.tex.bak new file mode 100644 index 0000000..0ba167f --- /dev/null +++ b/04/tw Pitagorasa/Twierdzenie Pitagorasa.tex.bak @@ -0,0 +1,64 @@ +\documentclass{beamer} +\usepackage[utf8]{inputenc} +\usepackage{polski} +\usepackage{graphicx} +\usepackage{tikz} +\usetheme{Hannover} +\newtheorem{twi}{Twierdzenie} +\title[Twierdzenie Pitagorasa]{Dowód twierdzenia Pitagorasa} +\author[Filip Kuczewski]{Filip Kuczewski} +\institute[UAM] + +\begin{document} +\frame{\titlepage} + +\begin{frame} +\begin{twi}[Twierdzenie Pitagorasa] +W każdym trójkącie prostokątnym suma długości przyprostokątnych jest równa kwadratowi długości przeciwprostokątnej tego trójkąta. Zgodnie z oznaczeniami na rysunku obok zachodzi tożsamość $a^2+b^2=c^2$. +\end{twi} +\begin{center} +\begin{tikzpicture}[scale=0.5] + \path [fill=blue] (0,0) rectangle (6,6); + \uncover<1>{\draw [help lines] (0,0) (6,6);} + \draw[ultra thin] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \draw[ultra thin] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \node at (3,3) {$c^2$}; + \node at (4,1.5) {c}; + \node[below] at (4,0) {a}; + \node [right] at (6,1) {b}; +\end{tikzpicture} +\end{center} +\end{frame} + +\begin{frame} +Dany jest trójkąt prostokątny o bokach długości a, b i c. Konstruujemy kwadrat o boku długości a+b w sposób ukazany po lewej stronie, a następnie z prawej. Z jednej strony pole kwadratu równe jest sumie pól czterech trójkątów prostokątnych i kwadratu zbudowanego na ich przeciwprostokątnych, z drugiej zaś równe jest ono sumie pól tych samych czterech trójkątów i dwóch mniejszych kwadratów zbudowanych na ich przyprostokątnych. Stąd wniosek, że pole kwadratu zbudowanego na przeciwprostokątnej jest równe sumie pól kwadratów zbudowanych na przyprostokątnych. +\begin{center} +\begin{tikzpicture}[scale=0.5] + \path [fill=blue] (0,0) rectangle (6,6); + \uncover<1>{\draw [help lines] (0,0) (6,6);} + \draw[thick] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \draw[thick] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (2,0) -- (0,4) -- (4,6) -- (6,2) -- (2,0); + \node at (3,3) {$c^2$}; + \node at (4,1.5) {c}; + \node[below] at (4,0) {a}; + \node [right] at (6,1) {b}; +\end{tikzpicture} +\begin{tikzpicture}[scale=0.5] + \path [fill=blue] (0,0) rectangle (6,6); + \uncover<1>{\draw [help lines] (0,0) (6,6);} + \draw[thick] (0,0) -- (4,2) -- (6,6); + \draw[thick] (0,2) -- (4,2) -- (4,6); + \draw[thick] (4,0) -- (4,2) -- (6,2); + \draw[thick] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0); + \path [fill=white] (0,2) -- (4,2) -- (4,6) -- (0,6) -- (0,2); + \path [fill=white] (4,0) -- (4,2) -- (6,2) -- (6,0) -- (4,0); + \node[below] at (2,4.5) {$a^2$}; + \node [right] at (4.5,1) {$b^2$}; +\end{tikzpicture} +\end{center} + +\end{frame} + +\end{document} \ No newline at end of file diff --git a/04/tw Pitagorasa/Twierdzenie Pitagorasa.toc b/04/tw Pitagorasa/Twierdzenie Pitagorasa.toc new file mode 100644 index 0000000..ae73954 --- /dev/null +++ b/04/tw Pitagorasa/Twierdzenie Pitagorasa.toc @@ -0,0 +1 @@ +\beamer@endinputifotherversion {3.36pt}