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document.write( "Two circles,centre A and B touch one another at C.Through C a straight line PCQ is drawn cutting the circles at P and Q.Prove that AP || BQ.
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document.write( "The plot is shown in the Figure.\r\n" );
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document.write( "Draw EF, the straight line perpendicular to AC at the point C.\r\n" );
document.write( "Then EF is a tangent line to the circle A.\r\n" );
document.write( "At the same time, DE is the perpendicular to BC at the point C and is a tangent line to the circle B.\r\n" );
document.write( "So, the line BCA is a straight line.\r\n" );
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document.write( "The angles ACP and BCQ are congruent as they are vertical angles. ( See the lesson (*) )\r\n" );
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document.write( "The angles ECP and FCQ are congruent due to the same reason. ( See the lesson (*) )\r\n" );
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document.write( "Therefore, the minor arcs CP and CQ have the same measures. ( See the lesson (**) )\r\n" );
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document.write( "It implies that the central angles PAC and QBC are congruent.\r\n" );
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document.write( "Thus the triangles ACP and BCQ have two pairs of congruent angles:\r\n" );
document.write( "(ACP and BCQ, as well as CAP and CBQ).\r\n" );
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document.write( "Hence, the angles APC and BQC are congruent as the pair of the third angles of triangles.\r\n" );
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document.write( "But these angles are alternate interior angles. ( See the lesson (***) )\r\n" );
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document.write( "It implies that the straight line AP is parallel to BQ: AP || BQ. ( See the lesson (***) )\r\n" );
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document.write( "QED.\r\n" );
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document.write( "The referenced lessons are \r\n" );
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document.write( " (*) Vertical angles\r\n" );
document.write( " (**) The angle between a chord and a tangent line to a circle \r\n" );
document.write( " (***) Parallel lines\r\n" );
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document.write( "Also, you have this free of charge online textbook on Geometry\r
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document.write( " GEOMETRY - YOUR ONLINE TEXTBOOK \r
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document.write( "in this site.\r
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document.write( "The referred lessons are the part if this textbook under the corresponding topics.\r
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