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Difference between revisions of "2007 iTest Problems/Problem 60"

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''The following problem is from the Ultimate Question of the [[2007 iTest]], where solving this problem required the answer of a previous problem.  When the problem is rewritten, the T-value is substituted.''
 
== Problem ==
 
== Problem ==
  
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Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>.  
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Triangle <math>ABC</math> has <math>AB=99</math> and <math>AC=120</math>. Point <math>D</math> is on <math>BC</math> so that <math>AD</math> bisects angle <math>BAC</math>. The circle through <math>A, B</math>, and <math>D</math> has center <math>O_1</math> and intersects line <math>AC</math> again at <math>B'</math>, and likewise the circle through <math>A, C</math>, and <math>D</math> has center <math>O_2</math> and intersects line <math>AB</math> again at <math>C'</math>. If the four points <math>B', C', O_1</math>, and <math>O_2</math> lie on a circle, find the length of <math>BC</math>.  
  
 
== Solution ==
 
== Solution ==

Latest revision as of 10:58, 16 April 2024

The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.

Problem

Triangle $ABC$ has $AB=99$ and $AC=120$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A, B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A, C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B', C', O_1$, and $O_2$ lie on a circle, find the length of $BC$.

Solution

111

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 59 		Followed by:
Problem TB1
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