Modern geometry . be two equicross ranges, and ifA A, BB, CC be concurrent, then dd must pass throughthe point of fig. 76. Let O be the point of concurrence of AA, BB, dd does not pass through O, let OD cut AB in {AB, CD} = {AB, CD} = {AB, CD}. A^. D[B^ _ AC. DB ** C^.aF ~ CB . AD . D^ _ DB ?• A^~A^ .*. D coincides with D, .. DD passes through O. CROSS-BATIO 131 Note. This theorem and Theorem 51 could be stated astheorem and converse. It must be carefully noted that it isgenerally not true that, if {ABCD} = {ABCD}, then AA, BB, CC,DD are concurrent. Ex. 535. Examine
Modern geometry . be two equicross ranges, and ifA A, BB, CC be concurrent, then dd must pass throughthe point of fig. 76. Let O be the point of concurrence of AA, BB, dd does not pass through O, let OD cut AB in {AB, CD} = {AB, CD} = {AB, CD}. A^. D[B^ _ AC. DB ** C^.aF ~ CB . AD . D^ _ DB ?• A^~A^ .*. D coincides with D, .. DD passes through O. CROSS-BATIO 131 Note. This theorem and Theorem 51 could be stated astheorem and converse. It must be carefully noted that it isgenerally not true that, if {ABCD} = {ABCD}, then AA, BB, CC,DD are concurrent. Ex. 535. Examine the paiticular case in which {ABCD}, {ABCD}are similar. Ex. 536. Place two similar ranges {ABCD}, {ABCD} in such aposition that AA, BB, CC, DD ore not concurrent. Theoeem 54. If two equioross ranges {PXYZ}, {pxyz} have a pointp in common, then xx, YY, zz are concurrent.
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