document.write( "Question 565710: O is the centre of the circumcircle of 4ABC, and AP is the
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Algebra.Com's Answer #366001 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "I will just explain the proof.  You must write it up \r\n" );
document.write( "in statements the way you were taught.\r\n" );
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document.write( "There are two cases, \r\n" );
document.write( "1. when BC is not a diameter, and \r\n" );
document.write( "2. when BC is a diameter. \r\n" );
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document.write( "Case 1:  BC is not a diameter.\r\n" );
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document.write( "Draw radii OC and OB\r\n" );
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document.write( "We use this theorem:\r\n" );
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document.write( "If a central angle and an inscribed angle of a circle subtend the same arc\r\n" );
document.write( "or equal arcs, then the central angle is twice the inscribed angle.\r\n" );
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document.write( "Central angle COP and inscribed angle CAP subtend the same arc CP; therefore\r\n" );
document.write( "the central angle COP is twice the inscribed angle CAP.\r\n" );
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document.write( "Similarly, central angle BOP and inscribed angle BAP subtend the same \r\n" );
document.write( "arc BP; therefore the central angle BOP is twice the inscribed angle BAP.\r\n" );
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document.write( "Since inscribed angle CAP = inscribed angle BAP, angles COP and BOP are\r\n" );
document.write( "equal.  Therefore OP bisects angle BOC.  \r\n" );
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document.write( "Triangle BOC is isosceles because its legs are radii.\r\n" );
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document.write( "Therefore OP is perpendicular to BC because the angle bisector of the\r\n" );
document.write( "vertex angle of an isosceles triangle is perpendicular to its base.\r\n" );
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document.write( "--------------------\r\n" );
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document.write( "Case 2: BC is a diameter:\r\n" );
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document.write( "Angle BAC is a right angle because it is inscribed in a semicircle.\r\n" );
document.write( "Since AP bisects angle BAC, it is 45°.    \r\n" );
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document.write( "We use the same theorem we used for the other case.\r\n" );
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document.write( "If a central angle and an inscribed angle of a circle subtend the same arc\r\n" );
document.write( "or equal arcs, then the central angle is twice the inscribed angle.\r\n" );
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document.write( "45° inscribed angle CAP and central angle COP subtend the same arc CP, so \r\n" );
document.write( "angle COP is twice 45° or 90°, a right angle, which is the same as saying \r\n" );
document.write( "OP is perpendicular to BC.\r\n" );
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document.write( "Edwin

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