Twierdzenie 1
Jeżeli
przez punkt P, którego odległość od środka danego okręgu jest większa niż
promień, poprowadzimy styczną do okręgu w punkcie A i sieczną przecinającą
okrąg w punktach B i C, to
![](data:image/png;base64,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)
Załóżmy, że:
Dany jest okrąg o(O,r), gdzie |OP|>r i A jest punktem
styczności, a B i C to punkty wspólne prostej przechodzącej przez punkt P i
okręgu
Teza: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAUCAMAAABxjAnBAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjqQZrbbZrb/kDoAkDpmkGY6kLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///b6suHagAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABgUlEQVRIS+2U21bCMBBFk3qhihaqUC+pFGhC8/9fyMzkQhqGZV9cutQ+QZMze+acASF+5rMr5WP/Da2Zh/5Qr78IDMWhspLw+OFMiSz3XtiXDQfOVCQvQOZVQZIXt69wb+FuubMBBjvUSxIomYC3/MCZCuW2vdrw4Fh8qKveNssUbO5hMDVDrl7AUWho+8Yb7cBRRR/0JXC4ZptbUHXrFNzBq4EmBm9VBMO8mh3ZgaNKQ8v7EmSs1eGaLt7dGInVwLIKOqaGItiUEAmBbYMpYpBO68MLKsz4Bk/4jP01hQOPwUMNyvkHhv3cO8uzEpnjdBpVtoGIqSwHDtcw3Rxs7rwJbr0ngqOKPmDEJ7Ca0RLQ+vriuGM5GFM4mUDfxr2zVkeVpqUZgfNyaOY52Da0zxglNtUypjFWR5XoQL7D3eKsTopD3f08+IKTdVKS/baVVS+0LFZTMg4qYXBDrkHDguM1MTzBclYxAfcPdfZMWK7PVReKJD+nvwRmff5/+YscOALTWTWV3TdmIAAAAABJRU5ErkJggg==)
Dowód:
Przyjrzyjmy się trójkątom APB i APC. Otrzymujemy:
, bo kąt wpisany BCA ma taką samą miarę jak
kąt dopisany PAB, oparty na tym samym łuku
Kąt APB jest wspólnym kątem trójkątów APB i APC, zatem na
mocy cechy kkk podobieństwa trójkątów, trójkąty te są podobne.
Zatem:
, skąd
co kończy dowód.
Twierdzenie 2
Jeśli
dwie proste przecinają okrąg odpowiednio w punktach A i B oraz C i D, a także
przecinają się w punkcie P, którego odległość od środka danego okręgu jest
większa niż promień to ![](data:image/png;base64,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)
Twierdzenie to wynika z poprzedniego twierdzenia.
Twierdzenie 3
Jeśli
cięciwy AB i CD okręgu przecinają się w punkcie P, to ![](data:image/png;base64,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)
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Przykład 1
![](data:image/png;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/wAALCAD3AcQBAREA/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/9oACAEBAAA/APZqKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKjnnhtommnlSKNeruwAH41DZ3j3m9xbSRQceW8g2mT1O08gfXB9qtUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUVnzamzzPbafD9pmQ7XfOIoj6M3r7DJ+lEOlhphc38v2u4Byu4YjjP8AsJzj6nJ960KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKq3mo21jtWVi0r/wCrhjG6R/ovU1VFpfalzqDG2tz/AMusL5Lf77jn8F/M1owwxW8SwwxrHGgwqIMAD2FPooooooooooooooooooooooooooooooooooooooooooooooooooopskiRRtJI6oijJZjgCs37Zealgaev2e3PP2qZM7h/sJnP4nj2NWbPTrex3NGGeV/vzStud/qT29untVuiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiis+41RfOa1sYxd3SnDKrYSL/fbnb9OT7UkelmeRZ9TkF3Kpyse3EUR/wBle59zk/StGiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiq93fW9igad8FjhEUFmc+iqOTVPydQ1Pm5LWNsf+WEbfvXH+04Py/Rfzq7Glrp9ukSCOCJeFGQBTP7SsAdv263z6eav+NTRzwzf6qVH/AN1gakoooooooooooooooooooooooooooooooooooooooooooooooornvFcuoyLb2WlLK1yd1wfLl8vhOgJ7gsVBHcZrZs72K806G+U7YpYhJluNoIzz6YrFk8ULqEjW+gRm9Kna9xGu6MH0B4BPuSB9elSWmlal5hneWG3lcfNMw8+cj03EBVHsFxVw6HDJk3V3e3OezzlR+SbR+lPj0PSYuV062JPVmjDE/ial/szT8Y+w22P+uS/4VHJomlS436banHQ+SoI/HFRf2Dax/wDHtNd2pzn91cNj8iSP0o8jWLYkxXkN4nZLiPY2P99eP/HaF1uOJ1j1G3lsHJwGl5jP0ccfng1pAggEHIPQiloooooooooooooooooooooooooooooooooooooooooooorI1XxBBYSrZW0bX2pSj93aQYLf7znoi+5x7ZrOsvDmqaraRHxhfJdupz9jtMx2/wDwPu56dePbvXSwwxW8SwwxpFGgwqIuAB7Cn0UUUUUUjKrqVZQykYIIyDWY+kyWh8zSJhbnqbd8mF/w6r9V/I1NZ6ms8xtbiJrW8UZMLnO4eqN0YfTkdwKvUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUVDdXdvZQ+bcSCNc4HGSx7AAck+wrDvL3VNTuDY2Sm0yoZifvxqejOR93PZR8x9VrS0jRbPRoGS3TdLId007cvK3qTWhRRRRRRRRRRVa9sbe/hEc6nKncjqcNG3ZlPY1Utry4sriOx1Ng5kO2C6AwJf9lh/C/wCh7elalFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFIzBVLMQABkk9qzW1Ka+Yx6UiSJnDXbn90v+7/AHz16ce9Vp4fsc6RwN9r1i4U7Z5gD5SfxPgcKoz0GMnA561p2NlFYW/lRlmZjukkblpGPVifWrNFFFFFFFFFFFFQ3VrDeWz29wgeNxgj+RHofeqVhczW9wdMvnLzKC0ExGPPQf8Asw7j6Hvxp0UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUVSvNTitZBAiPcXTDKwRDLfUnoo9zgVCNNnvmEmqyhkzlbSIkRj/AHv7/brx7VcurmHT7KS4k+SKFMkKOw6AD9AKraVazRpJd3YH2y6O6QZz5Y/hQewH5kk960KKKKKKKKKKKKKKKqalZG+tdsb+VPGwkglx9xx0P07EdwTS6dei/tFlKeXKpKSx5z5bjhl/P9MVaoooooooooooooooooooooooooooooopMjIGeT0paKhubu3soTNczJFGP4nOPw9z7VS3X+p/6vfYWp/jIHnP9ARhB7nJ9hVu0sraxjKW8QTccu3VnPqx6k+5qxWXc/8AEw1mKz6w2YFxNju+f3an6YLfgtalFFFFFFFFFFFFFFFFZcgGn62kqjEOoHZJ6CVR8p/FQR+C1qUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUVzOpW903iy3n8kyoDEIR5JYKMt5h3j7mAQcfxcDnt01ZsmpyXLmHSoluGBw87HEUZ+v8Z9l+hIp9tpaRzC6upWu7rH+tfgL7KvRfw59Sav0UjMEUsxwAMk1meH8y6aL6RGWS+Y3DBjyA33R+C7RWpRRRRRRRRRRRRRRRRVPVrRr3TZoU4lxviP8AddeVP5gVLY3S3tjBdIMCaMPj0yOlT0UUUUUUUUUUUUUUUUUUUUUUUUUUUUVUvNRgs2WJt0s7jMcEQy7/AIdh7nA96r/YLnUPm1OQJD2tIW+Uj/bbq304H161oxxpFGscaKiKMBVGAPwp1FFZ2vMw0eeNDh59sC/V2C/1q+iLGioowqjAHoKdRRRRRRRRRRRRRRRRRWZouYlvLQ8fZ7pwo/2Ww4/R606KKKKKKKKKKKKKKKKKKKKKKKKKKKjnuIbWFpriVIo16u7AAfjWf59/qfFqrWNsf+W8i/vHH+yhHy/Vvyq5Z2NvYqwgQhnOXdiWZz6ljyasUUUUVm6t89xpsPZ7sMfoqs38wK0qKKKKKKKKKKKKKKKKKKzbUeX4gv07SRQyAe/zKf8A0EVpUUUUUUUUUUUUUUUUUUUUUUUUUUVnzamZJXttOj+03CnDt0jiP+03/soyfp1og0sGZbm/lN3cA5XcMRxn/YTkD68n3rQoooooorM1HP8Aa+k+nmyf+i2rToooooooorHubC303Ur3xK9xduy2ZR7fzcxbV+bIX+9x1qWxv71r1bS/t4onlhM0ZhcsAAQCpyByMjnoa06KKKKKKKzEz/wlE3p9ijz/AN9vWnRRRRRRRRRRRRRRRRRRRRRRRRVW81G3sdqyMWlf/VwxjdJJ9F71V+yXuo837fZrc/8ALrC+Sw/23/ov5mtGGGK3iWGGNIo0GFRFwB9BT6KKKKKKKzdWOy402b+7dhT7BlZf5kVpUUUUUUUUUyWKOeF4ZUDxyKVZT0IPUVTsdIhsZjMJp55NgjRp33FE/ujj9Tyccmr9FFFFFFFZtr+88QX79o4oYx9fmY/+hCtKiiiiiiiiiiiiiiiiiiiiiimSSRwxtJK6oijJZjgD8azvtl5qfGnKbeA9bqaM5b/cQ4z2+Y8exq1Z6db2O5owzyv9+aVi7v8AUnt7dKtUUUUUUUUUVna+rf2NPKnL2+2dfqjB/wClX0YOiupyrDINOooooooooooooooooorN0YmVby7PP2i6cqfVVwg/RK0qKKKKKKKKKKKKKKKKKKKKKoT6ovnNa2UYu7lThlVsJF/vtzj6cn2pselefIs+pyC7kXlYyuIoz/sr3Pucn6Vo0UUUUUUUUUUUjKHUqwyrDBB71naE5WxNm7ZksnNu2euB90/ipU/jWlRRRRRRRRRRRRRRRRVPVrprPTJ5Y8+bt2xADJLnhR+ZFP0+0XT9Ot7NWLCCJU3HvgYzVmiiiiiiiiiiiiiiiiiiiq95fW9igad8FjhEUFnc+iqOT+FUxDfan81wzWVqf+WCNiVx/tMPu/RefftV+3t4bWFYbeNY40GAqjAFS0UUUUUUUUUUUUVl3RGn6xFeE7YbsLby+gfP7tvxyV/Fa1KKKKKKKKKKKKKKKKKy5f8AiYa5HEOYNP8A3knoZWHyj8FJP/AlrUoooooooooooooooooorLu9fgs7yW3kt7hlgEZmmULsTeSFzlsnkdgaZr0+vxfZF0KztrjfJi5aeTb5aeqjuar2kOo2khm/sdZ7hhhria8DOR6Z28D2GB7Vc+26x/0B4/8AwMH/AMTR9t1j/oDx/wDgYP8A4mj7brH/AEB4/wDwMH/xNH23WP8AoDx/+Bg/+Jo+26x/0B4//Awf/E0fbdY/6A8f/gYP/iaPtusf9AeP/wADB/8AE0fbdY/6A8f/AIGD/wCJo+26x/0B4/8AwMH/AMTR9t1j/oDx/wDgYP8A4mj7brH/AEB4/wDwMH/xNH23WP8AoDx/+Bg/+Jo+26x/0B4//Awf/E0fbdY/6A8f/gYP/iaPtusf9AeP/wADB/8AE0fbdY/6A8f/AIGD/wCJqG7fVL20ltptFiZJFwR9sH5j5eveqOhav4kltDbanpFuNQtztmCXYAYfwsODwR+oPTpWp9t1j/oDx/8AgYP/AImj7brH/QHj/wDAwf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Oblicz długość |BC|, wiedząc, że |PA|=6cm i |PB|=4cm.
Wiemy, że
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Odp. |BC| ma długość 5cm.
Zadania do zrobienia
1. Z punktu
poprowadzono styczną do
okręgu w punkie
oraz sieczną, przecinająca okrąg w punktach
i
, jak na rysunku
poniżej. Wiedząc, że
, oblicz
.
![](data:image/png;base64,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)
Odp. ![](data:image/png;base64,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)
2. Przez punkt
poprowadzono styczną do okręgu w punkcie
i sieczną okręgu, przecinającą ten okrąg w
punktach
i
(zobacz rysunek do
zadania 5.155). Wykaż, że jeśli
, to
.